Do deer find the leaves a treat? My guess, yes. There isn't much they don't find tasty around here...
What 4 legged creatures like fig trees, leaves and fruit? How bout woodchucks?
A friend of mine's dog ate his fig tree (it was small). It started with just a fig...then later he went after the whole thing.
We don't have deer...but I have observed rabbits around my trees. They eat everything else, including citris, but won't touch the fig trees.