if they are going to ban you, you will be banned when signing onto live, not just playing a multiplayer game. you can be banned from xbox live silver if you have a modded console, which doesn't even allow you to play multiplayer.
if you are banned from xbox live because of a modded console, you are banned from signing onto the network. that means you can't donwload any demos or anything because you cannot connect.
the only other "banning" that I have heard about is in Halo 2 they were banning specific gamertags from entering the matchmaking portion of it only. But that was for cheating.
when you are banned with a modded console, the actual physical console is banned. your account is not banned at all, you could put your hdd on another xbox360 and sign onto live no problem.
but keep in mind, some people with modded 360's aren't even banned so just because you have a modded 360 doesn't mean you will be banned when you sign into live.