Obviously, no one's going to ban domes, but there is some evidence that dome teams have an advantage in the regular season. You can read the WSJ article for the statistical details, but it basically boils down to this: Dome teams can be dominant in the regular season, and most of the best scoring offenses in NFL history have been dome teams. When the playoffs roll around, however, you probably don't want to be a dome team.
Of course, two of the worst teams in the NFL, the Rams and Lions, play indoors, so it doesn't work out for everyone. In order for it to be an advantage, you've got to build your team for it: speed, precision and timing, particularly in the passing game. See the Indianapolis Colts and New Orleans Saints for examples. Depending on the building, crowd noise can be a huge factor, too.
The bad news is that eventually, you're going to have to venture out of that dome, and your team that's built for a perfect surface, 70 degrees, and immunity to weather might have to play in Pittsburgh in January. It will be cold, precipitating, and if you attempt to cut, a chunk of Heinz Field roughly the size of a cheerleader will come loose. Best of luck, dome team.
Anyway, it's a fun philosophical football discussion to have: If you were an owner, would you build a dome, or would you play outdoors? There are a lot of factors. You know, a lot of ins, a lot of outs, a lot of what-have-yous. If you ever have that discussion with me, I'll be building my fake team in an outdoor stadium. I don't care if it's in unexplored Yukon territory, I'm not doing anything that puts me at a disadvantage in the playoffs.
Of course, none of that matters, because none of that has anything to do with why domes really exist, which is money. You'll draw more fans in Detroit or Minnesota if you're not asking them to spend four hours outdoors when it's four degrees. You'll draw more fans in Arizona if you don't insist that your fans roast themselves for four hours. Domes are also easier to sell to cities and taxpayers, as they're more likely able to be used for things other than football.