I'm from British Columbia; lots of BC is technically a rain forest, which pretty much means you're always starting your fire with wet wood.

The trick to getting wet wood to light is to generate a lot of heat when you first start your fire - that means using lots of extra kindling. Cut triple or quadruple the amount of fine kindling and build yourself a thick teepee with a good amount of tinder to get it started. Regardless of how wet a log feels on the outside, if you split it open, the centre will be relatively dry, so use that for your tinder and smaller kindling.

When you build your fire with lots of wood it's easy to smother your tinder, you need good airflow with wet wood, so make sure it can breathe. Your goal is to get a lot of flame as soon as you light it; this flash of heat will help you get some hot coals quicker, which will dry out the rest of the wood. Completely build your fire before you light it; this helps the fire grow faster once it's lit. You risk your fire burning out if you try to light your tinder first then build it up after it's lit.

To help keep your fire going, stack your fuel wood around the fire like a wind screen; they'll catch a lot of the heat off of your fire and dry out a bit before you need to use them.

Since you didn't limit the question to materials found in nature:

I've lit wet firewood in the rain using pieces of waxed cardboard. They burn very fast and hot. And wax firestarters are essentially waterproof themselves, so they're pretty reliable even in wet conditions.

You used to be able to get waxed cardboard from any supermarket, in the form of broccoli boxes, but it's getting harder to find now that more stores are using reusable shipping containers. Of course, you can always melt a candle and dip pieces of cardboard in it. I've also heard of molding wax and sawdust firestarters in a cardboard egg carton. You can take the whole carton with you and tear pieces off as needed.

Look for a small drier to dry stick. Take knife & cut upward into it so tou have feathers on it. Sick bottom in ground. upright light bottom feathers. Have small light tender handy. This will give you about 15 seconds of flame. Build fire up from there. between 2 larger small logs. once fire is going place 2 to 3 small logs above it. Dry as you can find. Leave room to keep your fire going. heat drys out the top logs & side logs. Pine is best as has oil in it. You need to look at sheltered sticks were rain don't hit to bad. for your starter stick. Stick about as big around as little finger is best. You may need to shave bark off it first.

Since you mentioned pre-cut firewood, I'm assuming you're car camping or at some other location where you can easily bring some store-bought materials (not backpacking).

By far the easiest way I've found is to use a synthetic log such as Duraflame for about $2. Place that at the bottom of the wood pyre, arrange the wet logs on top, then light the synthetic log.

The synthetic log will burn on its own for 2-4 hours (depending on brand and cost), well more than enough to turn a stack of wet wood into a roaring fire. No need to find or create kindling. You can use full size wood logs, sit back and relax.

Obviously if there is a very strong wind or it's actively raining, you will need to partially block those elements to get the synthetic log started. If the paper shroud of the synthetic log has burned up but the fire went out, use a stick or a knife to chip off and crumble about a 1" cube chunk of the synthetic log. That material will light easily and can be used to get the solid part of the synthetic log lit. Or carefully use an accelerant such as lighter fluid or gasoline.

even wood that's been kept covered is often very wet and can be difficult to light.

I will second that. Some people claim their wood is dry, but if you go look at how they "cover" it, sometimes you see that the wood is only protected from the lightest of rains in the lightest of winds. I have seen some "coverings" that any amount of normal wind would blow the rain right under and onto the wood.

Techniques for fire starting in wet conditions

What are some techniques for lighting wet firewood?

Get at the drier wood on the inside

Try to dry out the firewood as best you can, if that is possible. Either way, make sure that you cut into the wood; as long as the wood is only wet on the surface, you can get drier wood on the inside.

To get at the drier wood, you can use a bladed tool (generally knife or hatchet) to cut off the wet outer part. Also make sure you split the wood as much as you are willing to do; the increased surface area will help a lot toward getting the wood to catch fire.

Use lots of tinder and kindling; as much as you can make

Even more importantly is to use a lot of tinder and kindling. When I light in wet conditions, sometimes I spend most of my time making a pile of wood shavings with my knife. I make enough of them that I have a reasonably large pile to light, and plenty more to keep adding if the larger pieces are having a hard time catching. Have extra tinder on hand for the same reason, in case the shavings themselves are too moist, or in case the fire goes out and you need to re-start it.

On top of your tiny kindling, keep working your way up to slightly larger pieces; do not suddenly jump from small kindling to very large pieces of wood.

Some methods for more extreme wetness

If you are trying to start a fire while it is currently raining, as opposed to after a rain, it can be even more difficult. I have found that the humidity and rain are sometimes bad enough that no amount of kindling will do: the tinder and kindling are becoming too wet simply by being in the wet air. When your resources are this wet, you need to use something that catches easily and burns hotter, like fat-wood, magnesium, or super-fine steel wool. You can even combine these things: my father likes to use magnesium and fat-wood together.

Fat wood

Fat-wood is a naturally occurring kind of wood which is saturated with resin. You can also make a home-made variant by soaking wood in flammable resins. It catches a spark or flame relatively easily, and it burns hotter and longer than normal wood. You then continue as before, working your way up with larger pieces of wood, once this has helped you get your start.

Take some very small shavings off of it, and some larger shavings. You can even make toothpick-sized kindling and bigger if you're having a hard enough time. Usually, you make tinder and kindling from it then put what's left of your fat-wood piece away to help with more fires later, but if you need the fire fast or bad enough, or if it's a short trip or have plenty to spare, you can put what's left of your fat-wood stick in over your fat-wood tinder and kindling since whole fat-wood sticks help too.

Magnesium

Magnesium burns super-hot; many thousands of degrees. The magnesium for fire starting comes in a small block, and you scrape some of it off the block to make a small pile of scrapings/shavings. Then, usually a spark is applied by using a ferro rod; as long as the spark lands directly on the tiny magnesium shavings it should give you a super hot spot for a very brief time which you use to get your damp tinder and kindling lit.

Steel wool

Steel wool, the super-fine grade stuff, burns quite hot too. I once saw someone completely submerge it under water, take it out and shake it off for a minute, then drop a spark on it with a ferro rod. He then blew on it for a few seconds, and it was hot enough that kindling on it starting catching.

Charcoal

When conditions are very poor, my preferred method is to use charcoal. If I can get even a tiny spot on the charcoal to glow red, I can then blow on it to increase the size and heat of the red spot, then I hold a second piece of charcoal up to it so that it too can start getting hot; once the two of them are working together, it is easy to keep working up from there, possibly with even more small pieces of charcoal if the conditions are bad enough.

This technique requires a lot of blowing. Be prepared to be exhausted after. Also, once it starts to smoke, try to turn away quickly to breathe, and be prepared to accidentally inhale some smoke that will choke you out for a moment.

The up-side to this technique is that the charcoal burns hot, and you can keep forcing it to stay hot by continually blowing on it; so much so, that I have multiple times skipped kindling entirely and jumped straight from small pieces of charcoal to larger pieces of wood. Of course, lots of kindling is still helpful, and it will save some of your breathe. When all else fails, I get the initial charcoal red spot by applying a bic lighter to a sharp corner on the piece of charcoal.

Counter intuitive method: use green wood

This next one sounds completely counter-intuitive, but I have used it successfully. This one is back to wet conditions after a rain, not during the rain, and it only works if you have a nice, sunny day after the rain.

Putting objects out under a bright sun, especially if it's mid-day, can help dry things out a lot better. Smaller, thinner objects are going to dry out even faster, and this is good for your tinder.

Grab up a thin, live (yes, still green) branch from a tree. Scrape the bark away (if it is a kind of bark that is good for starting fires, then obviously use it as well), then start scraping at the inner wood. Try to keep the stuff you scrape off the wood to be as thin as you can possibly make it. I find it works best for me if I hold my knife perpendicular (straight up and down) on the wood and just scrape side to side, instead of whittling into the wood. Sometimes I even use the 90-degree corner on the back of my knife instead of using the actual blade, and I scrape this back and forth along the inner wood of the thin branch. I do this for a long time, building up as large a pile as I can.

These ultra-thin shavings are so thin that, by the time I am done, the first half of what I worked on is much drier. But now it's time to make them even better yet; go spread this tinder bundle out in the direct sunlight for 10 to 30 minutes to dry them out even more. Because they are so super thin, they dry out very fast in the sunlight. That is why I use green wood instead of dead wood for this part; I have found that I can get the shavings even thinner than I can with dead wood. Once it dries out in the sun for a quarter hour or half hour, now it's dry wood and it's super thin which is great for tinder.

With green wood, I can get the stuff I scrape off so thin that sometimes it looks fuzzy instead of woody. And since it is so thin, which is good for lighting, I have a few times gotten it to light after a rain even without the sun-drying, though don't skip the sun step if it's shining.

Yes, this sounds crazy and counter-intuitive. I saw a guy do it on YouTube and thought "He must be crazy for choosing green wood," until he explained it and actually got a fire lit with it. I had to try it to make sure he wasn't pulling our leg, and it worked great!

Conclusion

Whatever you do, make sure you have multiple alternatives available to you. I keep multiple tools in my pocket-sized tinder box. It has a tiny piece of fat-wood, a few scraps of char cloth, a few tiny pieces of charcoal, a small chunk cut off of a magnesium block (spot reserved, it's not currently there), and a small ferro rod removed from its handle. Sometimes I have a tiny tinder-nest in there too from the afore-mentioned scrapings off a thin, green stick.

With all of those options, it is rare that I need anything else to get a fire started even in wet conditions.

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Your other question

Are there ways to tell wet pre-cut firewood from wood that's only damp on the outside?

Try to push into the wood. If it is only surface-wetness, you should not be able to push into it at all; the wood should feel completely firm and tough as usual. The wetter the wood is, the more you will be able to push into it.

For example, I just did a fire recently where I did use some very wet wood. I had some branches that I could push into about a quarter-inch, more in some spots. I scraped this wettest part away and burned the rest; it did not burn well, but the fire was already established, so it did eventually burn.

For wood which is wet, but not wet enough to feel soft at all, you could try looking for signs like fungus. Not all fungus will be large mushrooms; even smaller spots, which could look almost like mold (and might indeed be mold).

The location of the wood is another indicator. If the wood was flat against the ground, it is more likely to be wetter all the way through. If it is not flat, maybe a branch that is curved enough that part of it is held off the ground, or one side is propped against a tree or on a bush or another branch, or if the entire thing is caught up in something and not on the ground at all, etc.: since it has been kept off the ground, it is likely to be drier on the inside.

In such cases the kindling material found on the ground might be also wet: try searching for thin dry branches on the lower parts of trees. Especially the pine trees are "built" in a way that the lowest branches die and dry as the tree grows - the rest of the branches are usually thick enough to keep this bottom part dry, and as so, perfect to be used as kindling material.

This summer I went out canoe camping in some light rain that turned into a torrential downpour and wouldn't stop. When it finally did, there was a puddle in the fireplace and we had no dry wood. We had a little paper with us, and we got tinder from the inside of logs, and all those tricks, but nothing worked and we had a small child with us who was getting majorly unhappy.

We had already laid birch bark down over the puddle to give us a dry place to work, and we had a fair amount of wood of various sizes, and plenty of dry matches. We also had our stove fuel (Coleman fuel here in Canada.) Just a few teaspoons underneath the dry wood, a match (tossed in from a respectful distance) and that whooomph that anyone will recognize/remember from the time their Dad did the same thing, and we had a roaring fire in no time.

I wouldn't do it every time, but every ten years or so it's the right thing to do.