As it stands, the majority of the modes in "Call of Duty: Modern Warfare 2" multiplayer do not support Xbox Live's party chat option. In fact, the only ones that do are Free-For-All, No Party Team Deathmatch, Ground War and two of the 3rd person modes. If you're only interested in chatting with friends while playing the other modes, you'll have to manually mute everyone else in your game (including any newcomers), or do this:
Step One: Go to Xbox.com and log in to your account
Step Two: From your profile page, click on Privacy Settings (it's just to the right of your avatar)
Step Three: The top option should say Voice and Text. Set it to Friends Only. Save.
Now whenever you play "Modern Warfare 2" multiplayer you'll only hear people you are friends with.
Obviously there are several downsides to this, compared to Party Chat. For example, if only half the party ends up in a multiplayer match (something that will happen from time to time), coordinating to get everyone back to the lobby is much more difficult. Also, if you find that you need to communicate with the rest of the team for some reason, you'd have to back up to the dashboard or log on to Xbox.com to change your settings again. Just switching over from Party Chat to Game Chat would be much easier.
The whole Party Chat ban thing feels rather arbitrary in the long run. I can understand Infinity Ward wanting to keep certain modes free of cheaters (Search & Destroy, Hardcore), but what makes 3rd Person Team Deathmatch more appropriate for Party Chat than regular Team Deathmatch? It's a total mystery for me.