Suggestion:
yes they can, ive had 2 consoles banned. I learnt that the hard way soon as mw2 came out, and they banned me on release date as they did with thousands of other people.
From personal experience microsoft will always know if a console is modded, but they wont ban anyone til a major game hit the shelf. Its there way of making money, so you will run out and buy another console straight away.
Id dont bother with it anymore, ive wasted to much money.

{ 2 comments… read them below or add one }
Yes.
They can review all of your game history connected to your gamertag.
One more reason not to pirate games, after all it is illegal and it is stealing.
Yes, they can. If it shows up on your online profile, MS can review it at any time, and since you have to mod the console to play it, they can always detect that. You could create an offline profile… but I suspect that would defeat the purpose of getting the achievements to begin with.
Incidentally… you cannot burn a 360 game to a CD. There's nowhere near enough space. You'd have to use a DVD. You'd also have to flash you 360 to play it, which voids all warranties. In general, it's really not a wise thing to do unless you can afford a new 360.