What's the difference between V-るうちに and V-ているうちに?

Very hard question.

For the first example, {食べる／食べている} うちに飽きてくる sounds exactly the same to me, especially when you intend to mean "you get bored of it after a while (in a single event of eating that food)". If you want to say that "you get bored of it when you eat it too much (repeatedly eating the food multiple times in your life)", I think the former fits better, since the latter has a stronger emphasis on the "while eating" aspect.

For the second example, I think only the first one works. 現実の非情さを知るうちに諦め... It is still possible to say 現実の非情さを知っていくうちに諦め... which sounds a bit more natural to me than the former. But the difference is very subtle.