Which kind of Xbox are you using?
The original (white) tends to have their drives go bad quick, the elite (black) has the problem less, and the kinect version has even fewer. The older your 360, the more likely the chance the drive just went bad.
It is strange, though, that it is only CoD. Have you tried using your disk in another 360? That would tell you if it's the disk. If you have tried that and the disk works on other 360s, there are some other things you can try before calling the drive busted:
1) try deleting the cache on your harddrive. You can do this by going to settings>memory>(select your hard drive)>clear cache. Sometimes some games can put files in the 360's cache and it may mess with other, usually newer, games.
2) try deleting and redownloading your gamertag profile; you'll need your email and password for the account incase you set those up years ago and forgot both (like I did
)
3) finally, try installing the game to your 360. [tangent]Most games made today allow you to install the game onto the hard drive so that files will load faster because the bandwidth in the hard drive is MUCH higher than a dvd drive; this is actually why many people are moving to downloadable games because games published on disks do not have the bandwidth the games need to play at a decent fps.[/tangent] I don't exactly remember how to install a game on the 360 from a disk, but I do remember from the 1 time I did it, its as simple as finding the button for it somewhere on the dashboard. A quick google search should yield that info.
As for the noise you hear, that is normal. Every time you close the disk drive, the 360 will try to see if there is actally a disk in the tray and it does this by simply running the drive, so the spinning and clicking you hear is expected. This noise is also louder if there isn't a disk in the drive.