The first year the wood we had was wet. Outside, in the rain, snow covered kind of wet. We ran the stove a lot with the damper open, and with the door propped open. One lesson learned was, that you can burn sloppy wet wood, and it won't hurt anything, if you keep the stack temp high, in this case 400-600deg. We couldn't get those temps any higher, because it just would not. We checked every month that the stainless chimney was staying clean. It was. We had no issues, other than that it produced marginal heat, was a bugger to get lit, difficult to stay lit, difficult to keep a consistent burn etc etc etc. We did it, didn't know better at the time, worked with what we had, and were determined to get some wood socked away for the future.
+1 on this. Lots of people will tell you not to bother, but it is highly dependent on particulars, including: i) do you have dry kindling? ii) can you do hot reloads much of the time and modify what your burn cycles look like? iii) what is the draft and airflow like on your stove? iv) what species of wood? (and possibly some other questions too). We burned about 2 cords of wood last winter, much less-than-ideally dry. No buildup issues says the sweep, almost every fire burned down clean to ash with no soot on the glass, and we got plenty of heat (about halved our oil usage relative to before the stove). We had enough dry wood to get fires started, were around mostly to do hot reloads, have a very strong draft and (for better or worse) no airflow control on our MF Nova, and a sizeable fraction of the wood was ash which burns pretty well even when not very seasoned (low green MC and ignites well). If you get a hot reload right with wet wood you might hear sizzling but you shouldn't see much smoke coming out of the chimney.
The energy penalty from burning wet wood is not very large, so long as you can get the wood to ignite, at least on its outer layers, rather quickly. Cellulose is about 16 MJ/kg of energy when you burn it cleanly. Water is 2.5 MJ/kg to vaporize. A 10% increase in moisture content is thus only about 0.25 MJ penalty, or around 1.5% of the heat you'll get out of the kg of bone dry wood. But, if the wood won't ignite, then you'll lose a much larger fraction than that as heat up the chimney while you've got the door and or damper wide open. If it smolders for a long time with a low stove temperature, you'll lose a lot more than 1.5% energy as unburned gases out the top of the stack (and risk creosote buildup). Getting wood to ignite quickly and not smolder in a cold stove is where a hot start with dry kindling, the wood species (as mentioned by other posters), as well as the hot reload, and probably strong airflow, all combine. You can get most of the heat out of the wood still, but you will need to modify your firecraft and may need to do frequent, small reloads on big beds of hot coals -- the ideal of running long combustion cycles with full reloads and low air for sure will not fly with very wet wood.
Drying wood for a few days near the stove -- as mentioned by many here -- will do a heck of a lot for drying the outer layers of the wood, which will make it ignite much more easily even if the interior is still quite wet.
Finally: don't recall if you have a stove with a catalyst but be careful about burning a lot of construction residue that may have galvanized nails in it.