Do you mean OTB? Or live, or online? If OTB, then it would have to be some event where you were signing up with rating on the line. If you mean live chess here on this site, then you can play unrated until the cows come home. The same goes for online chess, or correspondence chess (your turn, then my turn, 3 days allowed per move, or whatever time control was agreed upon.)
Can informal games alter elo rating?
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If you both sign a copy of the score sheets and a certified TD sends them in to the organization (USCF, FIDE, etc) for rating, then yes.
Otherwise, no.
Because this costs money, the TD would probably want you pay some money. So in effect it becomes a rated match... or two person tournament if you like.
Elo is a general term for any rating system that's based on the formula (which to my knowledge is all of them). So if you mean something less formal like a club rating, then sure, whatever games the club organizer chooses to count and put in his computer to re-calculate your rating will count.
At one club I'm rated over 2400, how about that! (Of course no one's rated below ~2200, but you can just ignore that).
If 2 rated players play outside of a "real" tournament, but with time controls, can the results of the game be used to alter their official chess ratings?
I don't even have a rating, but I was just curious.