Is there any danger in chopping down every tree I see?

I recently cleared my farm of all trees and rocks and personally kept the seeds. But if I just so happened to chop down every non-fruit bearing tree in the whole town and also sold/destroyed all of their seeds does that have any negative effects on the game? And if in the future I wish to plant some more, will I be able to get maple seeds, pinecones, acorns from another source?

There's no danger, no. Trees outside your farm will regrow over time. (I haven't timed it, but my offhand impression is it takes about half a season or a bit more.)

This makes town and forest trees a renewable source of wood, sap, and tree seeds. Seeds being a renewable resource means you can always reforest your farm if desired.