In League of Legends Summoner's Rift (5v5), there are 3 symmetric lanes.
If only the minions play, does it last forever? Are the lanes fully symmetric?
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.
It's possible for minions only to finish an entire game on Summoner's Rift, it's been tried several times and recorded in videos such as this one (where the red team's minions win) or that one (where the blue team's minions win).
Basically, even if both nexus send the same amount of minions at the same time, due to placement, minions focusing or not the same opponent minion and other factors, a team of minions can take the lead and progress towards a tower and kill it. The game will continue like this until one of the inhibitor is destroyed, where at this point it's likely the minion's team which managed to do so will win.