I agree with the IAC, it gets oil in it from the Throttle body and can gum up. But another thing that could cause the issue would be a vacuum leak or fuel pressure.

I had an issue with a pickup pump that when it warmed up would not let the car idle, it would sort of surge up and down and then die. You could hold down the brake the give it a little gas and the car would run. Blamed the IAC and cleaned it and lubed it. But the issue did not get any better and after the car warmed up a bit every red light was an adventure. Jumpered the pumps with the engine off and listened and you could hear this whooshing or surging sound from the pumps, the sound would come and go and that sort of matched the issues with the idle. It was not the normal pump humm or buzz.

Changed the pick up pump and fixed the problem.

Hope this helps,

Paul