I don't have the computer that tells the individual tire pressures, but I do have the tire pressure monitoring system, and a light will come on and warn me if they get to a low pressure. I have yet to see the light come on (except when starting the vehicle. It is the yellow light under the D for Drive on the right instrument panel).
If the monitor is going off, and telling you the tires are low, and you manually check them and they are okay, then, yes they will diagnose and figure out what the problem is with your monitoring system. The tires themself are not covered under warranty because they are a normal wear and tear item, unless they are stated in the lifetime warranty (if you have that warranty - I was unlucky and purchased before that warranty came out).
I would contact your dealer by phone and ask them what is covered, and see if they can get it fixed, also see if they will get you a rental car/loaner while your Avenger is being fixed, that could be under your warranty as well.
Last time my Avenger was in the shop they got me a Nitro for 4 days as they tore apart my transmission.