When starting the car with the Autostart AS-2775 remote starter, the vehicle starts properly, but the engine stops 3 seconds later, and I cannot figure out why. The vehicle is a 2007 Toyota Corolla CE, bought used with the remote starter already installed. The other features work properly (unlock, lock, panic, etc). Anyone familiar with this problem?

It sounds like the Anti-Theft system no longer recognizes the AutoStart system.
If that was a dealer installed option, check with your Toyota dealer to see if they have anyone who is familiar with your system.