I never notice anything but I usually warm them up at least halfway.
That being said, I don't play online that much but when I do, I hardly ever warm the relievers. Nothing seems to happen. The only reason I don't warm them up is because it won't actually stop them from warming when you click on "ready and waiting".
Unrelated:
Has anyone noticed on recent moments, it won't let you access the quick menu to go to the bullpen? You have to pause and go to manager submenu.