Why are my dwarves not gathering wood?

There is one possible reason not mentioned above. In your dwarves' standing orders is an option disable all wood-hauling. The standing orders are where you can change settings like telling your dwarves whether to gather refuse, whether to auto-loom thread into cloth and so on.

o to get into the standing orders screen, then w to toggle wood-hauling on or off.

The hint that allows us to answer this question is that your dwarfs are specifically ignoring wood, rather than some other resource. You should check the orders menu. One of the orders available determines whether dwarfs bother stockpiling wood at all, so make sure that's set to "yes, they should." There may also be a setting in the orders-Forbid orders submenu causing everything produced outside to be automatically forbidden. I can't remember if there is such a setting (and I can't find it on the wiki), but if there is, it would affect wood and goblinite more than anything else.

A second cause has to do with individual labors. Wood has its own hauling labor category. All dwarfs start by default with all hauling labors turned on, but it's possible that in your zeal to keep dwarfs on task rather than pointlessly moving things around the fortress, you've inadvertently turned wood hauling off on all of your dwarfs. You can check this by viewing a dwarf, and going to preferences-labor, opening the Hauling submenu (with numpad +/-/Enter), and checking what is turned on or off. If you're using Dwarf Therapist (which you should!), the hauling labors are all pretty far off to the right of each row; make sure at least someone has most of them activated. Developed fortresses tend to require a lot of hauling, so it might be a good idea to set some of those useless unskilled Peasant immigrants to the "Mover of Things" job.

Another possible cause has to do with burrows. Wood, unlike most other resources, is usually found outside. If most of the dwarfs are not allowed to go outside, then they won't be able to collect the wood. The few dwarfs who are allowed to go outside (the woodcutters themselves) will eventually get around to it, but probably not until they run out of trees they've been told to murder. But if you're new to the game, you probably haven't been mucking around with burrows yet.

If you've accidentally assigned a stockpile to have a bin or barrel amount, the dwarves will not put wood in it. This is because they can't cram a log into a barrel.

Thing that worked for me was simply to remove wood storage area and designate new one. 20+ dwarves rushed then immediately to pick up the wood ;-)