If I close the Oblivion gates nearby the cities early in the MQ, will it just spare me the work when it's finally time to gather allies for Bruma or will it cause a glitch when that quest comes?

No because new gates open up for that quest, I believe.

Not exactly. The gates for that quest open after the "Dagon Shrine" sub-quest. You can close them at any point after that, and you'll receive credit for them when doing the Allies quest.