Someone who's illiberal is close-minded and intolerant. Your illiberal grandfather might constantly complain about "kids today with their long hair and tattoos."
An illiberal person tends to have old-fashioned values and opinions and isn't usually interested in considering other perspectives. It's illiberal to dismiss unfamiliar religious beliefs — or the lack of any religious belief — as just plain wrong. The original, 1500s meaning of illiberal was "ungentlemanly," from the Latin word illiberalis, "ungenerous, mean, or unworthy of a freeman." The "narrow-minded" sense of the word is from the mid-1600s.