I've spent some time googling this, but can't find exactly the same situation reported - though I'm sure it can't be unique. Wondering if anyone else is encountering this or has seen it discussed elsewhere or knows anything about an intended patch for it down the road?
Originally, the game had no real problems. The only issue I ran into was (from the very first time I put the game in), the title screen would hang. That is, the title card and then the whole green on black intro video (and the demo that runs if you let the game sit on the title screen for awhile) that plays will hitch every few seconds - like a buffering youtube video. I put this down to some weird glitch they'd probably fix eventually, but it continues to this day. And, yes, the game is installed to the hard drive.
However, after a couple weeks, they released a patch and ever since then, I run into problems going to the multiplayer menus, too. Now - when I start the game and select "XBOX LIVE", it hangs... then the screen turns black. Then the game starts all over again back at the main multiplayer menu and I have to select "XBOX LIVE" again - and the second time, it actually takes me into the multiplayer options. It didn't do this until after the patch and it's annoying enough that I sometimes just don't play MW3.