Joker is the term that describes people who joke. Are there any terms to describe the opposite of joker?
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A formal adjective is
Some informal nouns are
I can't think of any formal nouns or informal adjectives.
If you want to go the metaphorical route, there's "machine," "robot," and such like. Of course the meaning changes depending on context, but still, it's an option.
The mechanical metaphor sort of implies a general lack of emotion, though, in which case "Vulcan" might be more obvious for some people.
Another slangy sort of option would be "stiff." In my experience it doesn't imply anything other than being unnecessarily formal, tense, no fun at all, which to me means not joking around and having fun with everyone else. Still not a perfect match, though. In certain circles being a "stiff" also implies being dead. You might want to keep that in mind just in case.
Some synonyms of already-suggested killjoy are grouch, spoilsport, and wet blanket. However, I'd look for synonyms of strait-laced, "Having narrow views on moral matters; prudish." (Prudish: "of excessive propriety; easily offended or shocked...") Synonyms of the latter include demure, priggish, prim, prissy, puritanical, square-toed, squeamish, straightlaced, straitlaced, tight-laced. Going for synonyms of recently-suggested deadpan gives impassive, poker-faced, unexpressive.
Edit: Prompted by Larry Morries comment on non-jokers I looked up some uses of it. It isn't a common term (fewer than 3000 Google hits) but in a half-dozen pages I looked at was consistently used specifically to contrast people who joke with people who don't. Example: 1898 J. F. Muirhead book, page 139.