People would not say "Thanks for your wishes." That's weird-sounding. Maybe if you were being sarcastic?
I wish you would die a painful death!
[rolling eyes] Thanks for your wishes.
If someone said "I wish I could be there with you so you don't have to go through that ordeal all by yourself." some natural responses would be:
Aw, thanks for saying so.
Thanks for thinking of me.
I appreciate the sentiment!
But "Thanks for your wishes." is stilted and incomplete-sounding. And "Thanks for your warm wishes." is also awkward, but a bit less so because it doesn't carry the sarcastic edge. (Similarly "Thanks for your kind words." is less weird than "Thanks for your words.")
Slightly more natural would be:
Thanks for the warm wishes.
Because it emphasizes that the wishes were given to you and not somehow still in possession of the giver. Think about getting a sweater for your birthday... you would not say "Thanks for your comfortable sweater." It's yours now, isn't it? You'd say "Thanks for the comfortable sweater (that you gave me)."
But if someone sends you feedback on something you might say:
Thanks for your thoughts.
Because the thoughts do still "belong" to them, so the "your" is fitting in this case.