Can tree shade hurt a neighbor's house?
If it were sunnier on his deck, he may have less rot as the dew would dry more quickly and completely. Will removing the tree eliminate the issue? Probably not completely given how crowded things look.

You're certainly not obliged to remove things that block sun from your neighbors property, however. Your house might also qualify as a sun-blocker, but I doubt you'd take that down.

Ultimately you'll have to feel out whats right and neighborly based on your comfort level and your relationship with your neighbor.