Homies have displayed pain, or some signs that might suggest it, in several occasions. Covering as many cases as I can think of, but with no pretense of listing everything:
- chapter 846: Kingbaum, while being slowly cleaved in half by Amande, describes his situation as unbearable pain and begs for death.
- chapter 849: a few animal-based homies scream in pain when they are hit in a fight (a somewhat obvious case).
- chapter 853: Prometheus laments that a seemingly minor injury received from Brook keeps hurting.
- chapter 890: Prometheus screams right before and right after being hit with water, and Zeus screams after he's been cut in half by Brook.
- chapter 1001: Prometheus screams after being cut in half by Zoro.
Homies in chapter 845/episode 809 do not seem to feel pain, so it's debatable whether they don't altogether or they do but don't show. They might be sharing with the croquembouche homies from chapter 826 a psychological mechanism that has them crave so strongly being eaten by a person who would praise their flavour, that it overrides their self-preservation instinct.
As for the reason why either of these options might be, it's the rabbit hole of how the Soru Soru no Mi works. It's apparent that some homies - chiefly the ones based on animals, the special homies made using Big Mom's soul and some strong regular ones like Kingbaum - have a deeper psychology, while most of the others have the bare minimum amount of sentience that enables them to perform their task. I find it obvious that much of it should depend on how Big Mom designs the homie upon its creation, although I can't rule out the possibility that limitations in the raw material play a role. I am unaware of a conclusive, detailed description of this mechanic, and I would refer you to fandom wiki https://onepiece.fandom.com/wiki/Homies for a few reasonable suggestions that may or may not agree with my opinion.