If no champions play, does the game last forever?

Solution 1:

No. Random variations in what targets the minions acquire will cause the game to eventually end. For example, if all 3 melee minions on the blue side target the same minion on the purple side, but each minion on the purple side targets a different blue minion, then the blue side will "win" that minion war by having one minion survive while the other side has no minions left. If this happens enough times, repeatedly, eventually minion waves alone will push towers, and ultimately, the nexus.

It can take a long time, but one team or the other will win. Which team wins, however, is random. Neither side has an inherent advantage if left to simulate itself.

This assumes that there are actually champions in the game but they are just idle at the fountain. If no one is in the game, it will simply end. It will not appear in the match history, and it'll be like the game never happened.