I have an RFID door opener. It works reliably month after month. The only time it failed was when we had a power outage and then a "brown-out" where I measured about half the normal line voltage. I think the board (temporarily) stopped working because the power company brought the power back up slowly. Possibly the RFID reader had failed and not the Arduino. However simply powering it off and on again fixed it.
I'm not sure if I had brown-out enabled on that particular board, possibly not.
If I was worried (I'm not because there are other ways into the house) I would make sure the brown-out was configured correctly, and make the processor turn the RFID reader on (via a MOSFET) in an orderly way. Also I would have a watchdog timer set up (which I haven't at present) to force a reboot if it gets into some loop.
I don't think there is any reason to suppose the the AVR chips are unreliable. In any case as Retrolefty said, I would be more worried that the RFID reader failed, or the door lock failed.
How long can it last until it skips a beat or stops working properly or downright fails?
Operating within design parameters, there is no reason for it to "wear out" or "skip a beat".