Zonies Rock! :shock:
Sounds like you have a smart cat.
IR could work, but you'd need to add some sort of power supply to the collar for your dog. I'm not sure how often you are willing to charge or replace the battery in the collar. It may end up being pretty expensive in the long run.
Also, considering IR oscillations, if you put your ear next to a TV remote control and push a button, you might be able to actually hear the oscillations as a faint, very high pitched tone. I can hear my remote. My point here is that I'm not sure if the dogs ears will be sensitive to these tones or not, but I think that it is something to consider / investigate before building a full blown IR system.
Maybe we could look at the option of preventing the problem from occurring in the first place. I'm assuming that the washer and dryer motors create a large-enough magnetic field to throw off the sensor of the dog door? Is there any option of shielding this sensor from this magnetic field with the dog collar still operational? I imagine that this might be pretty difficult to do, but I suppose the opportunity is there.
Is there the option of placing the magnetic sensor on the outside of the dog door? Maybe this will help prevent the disruption without sacraficing the functionality of the door?
Perhaps the simplest fix would be to lock the dog door while the washer and dryer are running?
Just tossing out ideas...