Well, yeah, I don't remember God's dialogue 100% but he said that Dog is inflicting pain upon himself to quiet down God.
So if he doesn't feel pain then it shouldn't have been able to quiet God down.
Of course, they're a Nightkin, so they "might" function differently, but I highly doubt that.
I think that in the old games when you shoot Unity members in the eyes the info-box says stuff like "You critically hit Super Mutant in the eyes for 34 damage inflicting heavy pain."
So I think they do.
Maybe the mutant in Tactics is different though.