Do I need to register my stay at my girlfriend's house in Thailand if I'm on a 30-day visa exempt entry?

I would suggest that, if it’s a relatively easy trip to the local immigration office, that you do have your girlfriend register a TM30 when you get there. I’ve been travelling to Chiang Rai regularly for the last ten years or so and previously never did a TM30. However, just last month my wife did register one at our local immigration office and, on the bus to the airport on my way back to Bangkok, it was checked by the police at a checkpoint. I’m not sure what would’ve happened if I hadn’t had one, but on all previous trips a quick flash of the ID page of the passport was enough. This time it got a thorough inspection. TM6 and TM30 both inspected.

I would highly doubt that she has registered to use the online TM30 system (because until late last year no one really did it but hotels, guest houses etc).

Here's the online system she can register to use and once she gets a user name and password she can then register you online;

okay, you're getting in on what's called a 30 day visa exempt entry :)

You don't fill out a TM30 on the plane :O you fill out what's called a TM6 Arrival/Departure card and that has nothing to do with the TM30 thing where you need to register with the immigration office within 24 hours of taking up residence.

With that being said

I would say IF you are not going to go to the immigration office for any service while you're here (extension of stay etc) that if she didn't file a TM30 for you no one would know ;)