Sid’s toys were clearly troubled by their experience, so it can be assumed they can either feel pain, or some form of mental pain when parts are removed.
I think this is kinda weird because in all TS movies the toys got their limbs yanked/ripped/broken off and did not show pain at all. They just reattached them.
Spoiler
In the new movie there is a plush toy who loses half of his body to a cat and you can see them all joke about it later on with the sentient half sitting around laughing.
With Sid they said he "tortures" toys. So maybe torture is more about disfiguring them. Like a custom gone wrong vs a custom gone right.