Lately, I've had this noise inside the car, sounds like its coming from behind the dash or the climate control system. I hear a lump, lump, lump, lump sound, even after I turn the car off and remove the key. I'm guessing its some kind of stepper motor for the climate control? Not sure...

 

I think you are very close.

Search for blend door stuff. We have had several good threads with part numbers for the blend door actuators and a Jaguar TSB about the same thing.

I recently tried to change the fresh air/recirculate door actuator on my 2005 STR. The part was made wrong and the strange thing is since I removed and messed with the actuator but did NOT replace it. The tap, tap noise is gone! Almost like a bit of exercise got it working again?

You might be able to tell which of the 4 actuators is making the noise if you can narrow down where in the dash area the noise is coming from. Mine was on the passenger side down low and that actuator is right behind the glove box.

When I get the right actuator I am not sure I want to change it since it seems to be working correctly now?
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Mine does the same thing. Once I turn the car off I can hear it.. Sounds to me like a gulping sound