I can't play MW3 or BO online but Xbox live works.

The Call of Duty servers were more than likely down while you were attempting to play. A good test to see if you are connected to live is to try and access the Marketplace from your Dashboard. If you are able to you are connected to Live.

If you are still having the issue, your NAT may be set to strict. If that is the case use the following guide:

support.xbox.com/.../nat-type-strict

Hope that helps!

I'm experiencing the same problem, as soon as I go back to dashboard it signs me back in to xbox live.  I am able to play Block Ops though so maybe you are experiencing a connection issue.  You might want to try some "hardware" solutions before trying anything else.  Try changing storage devices, changing game disks, and changing consoles before contacting xbox support.

If you find a solution to the problem, please message me. Right now I'm considering cancelling my current gamertag and registering a new one because I believe it's an issue involved with my gamer profile

Hope I was able to help!