It goes back to Greek, and then descends through Latin and French to English.
In Greek, the suffix was added to nouns to indicate connection with or belonging to. It could be used in a wide variety of ways; ὅπλα (armour) plus the suffix gives ὁπλίτης (a heavily-armed soldier). But the Greeks used it especially frequently for ethnic and local designations, and when the Hebrew scriptures were translated into Greek that gave us Israelite, Levite, Moabite and so forth.
In Christian use it served to identify sects or movements by reference to their founder or leader (Maronite, Puseyite, Hussite) , some belief or tenet (Monophysite) or some practice (eremite) and it also turns up as an identifier for non-religious ideological groups or movements (Williamite, Jabobite, Parnellite, Pre-raphaelite). As Desert Dumpster says, it’s not much used nowadays, and if it were used I think it would be vaguely denigratory or pejorative - if someone got called an Obamaite, for instance, I think we’d assume that the person so naming him had a somewhat negative view of his political position.
In another direction, “-ite” turns up a lot in mineralogy, used in huge variety of ways - graphite is so called because you can make marks with it, azurite is blue, we can guess who discovered brewsterite and where labradorite was found, etc. It turns up in chemistry, in naming salts of acids - nitrite is to nitrous acid as sulphite is to sulphurous acid.
As for naming nations and ethnic groups, in general any group that we have encountered since the classical Greek period has not been named with this suffix.