Would modding the Xbox 360 version effect the Xbox One version?

Meaning if I modded something in Gears of War, would it effect the Xbox One backwards compatibility version of it?

No, it won’t. It’s possible for modifications to carry over on a GameSave, but that’s if they allow you to carry over that GameSave from the 360 version of the game to the X1.

Exactly what Troy said, but for right now, Gears of War is not on the list of backwards compatible games, so for right now, you will have to stick to playing one of the 22 games that are available.