An alarm that beeps when the door is left open more than X minutes (say, 5 minutes) only requires a stupidly simple circuit and about $5 in parts.

No smarts needed (though it's probably cheaper to make it with a microcontroller than have the timer circuit be done with discrete parts).

Hmm, my fridge is in my kitchen, which is in the middle of the house. I've never been in a situation where the fridge door has been open more than a couple minutes without me noticing, and I have three kids.

I've had two fridges die (well, the same one twice), and that sucked, but there's not much I could do about it even if I knew a few hours earlier, and I use the fridge enough I'll notice within a few hours. Refrigerator deliveries often happen after a few days, and I'm not going to keep stocking ice during that period, I'll just consume what I can and move the important stuff to the mini fridge or chest freezer.

It would be cool, sure, but it's not worth having it connected to the internet.