I think I may have the solution, after having suffered through the same problem for months. I have a Kenmore Elite side-by-side which is really the same as KitchenAid and Whirlpool. Having thought of complicated solutions first, I have finally gotten back to step 1, simple solution. Tape the switch on the top of the freezer compartment and see if that works for you. I believe the door while seemingly closed is not pushing the switch in far enough. If that works for you, you can probably glue something to the top of freezer door so it can push in the door switch a little better. The reason that it beeps is that the switch is not pushed in enough and the light is still on inside while the door is closed. The refrigerator then thinks the door is still open. I really hope this works for you as I can sympathize with you having listened to the beeps for months. I feel sorry for people who have changed one circuit board after another to no avail. And there are so many stupid repairmen out there offering dumb advices and perform needless repairs.
dl, July 2009