How do I simulate a physical push button from an AVR
The closest thing to simulating a real push button would be to use a reed relay, such as these.  This particular one with a 5v coil voltage is available from Digi-Key for $1.10; there are others available with 3.3v coil voltages.

The advantage of using a reed relay, is you just wire the contacts in parallel with the original push button, and don't have to worry about how it fits in with the rest of circuit.

The following circuit can be used to drive the relay:



The wire on the left (going into the base resistor R2) comes from the AVR output pin.  The pull-down resistor R1 keeps the transistor off in case the output pin on the AVR is put into an high-impedance state (e.g. when the microcontroller is reset).