The wood's wet and won't burn - should be "the wood is wet and won't burn" While it is not technically impossible to make a contraction with "wood" and "is" in this sentence, readers would assume that "wood" is possessive and will get confused The wood was wet and didn't burn AND The wood was wet and wouldn't burn are both grammatically correct.

"the wood's wet" is incorrect for writing, because you are possibly possessing the wet wood, for the purpose of writing and following plural apostrophe conventions. But for speaking it is a perfectly natural contraction, to say "the woods wet" where the two words wood and wet slide into one word "words" which sounds like a plural. "the wood is/was wet and wouldn't didn't burn" = the clearest writing style to avoid confusion.

"The wood's wet" is a natural contraction and not ambiguous. The other two statements are grammatically correct but have slightly different meanings. The wood was wet and didn't burn. (sounds terse and factual, almost unnatural in daily speech) The wood was wet and wouldn't burn. (Has the sense that we tried several things to get the wood to burn, but were unsuccessful. You would be more likely to hear this phrase.)

oops, mean to type - The wood was wet and didn't burn.

The wood wad wet and didn't burn.