Update here friends.
I reinstalled the LNA, and after uploading a GPS test firmware everything worked flawlessly, GPS lock and cool LNA. Obviously, I was intrigued and thought I had made some stupid mistake.
I continued working on other sensors on the board, and then I noticed the LNA was suddenly hot again. The LNA only got hot when a different code was uploaded, and the code in question had a piece of the GPS test code that initialized the GPS module by enabling the ON_OFF pin (that activates both the GPS module and the LNA) and performed a reset via the NRST pin on the GPS module.
After an afternoon of debugging and burning the tips of my fingers touching the LNA to check if it was toasty, I noticed the problem occurred at a line of code where I disabled the 3V3 DC converter that feeds all the peripherals (done like that for power-saving reasons) and enabled the 3V3 power again AFTER I had initialized the GPS.
So the magic sequence to turn the LNA into a toaster is:
1) Have the 3V3 power off;
2) Enable the ON_OFF pin (HIGH signal from an STM32 - powered by a separate 3.0V LDO Linear Regulator);
3) Turn 3V3 power on.
4) You now have active thermal control
After that, the only thing that cools down the LNA is turning 3V3 off again. Disabling the ON_OFF pin does nothing.
Also, I thought it might be a transient in which the GPS module draws too much current, but in that case the LNA would only heat up momentarily, and in reality it stays hot for as long power is on.
Note that if I FIRST turn the 3V3 on and then turn the ON_OFF pin HIGH everything works fine. So yes, I can easily solve it in firmware and move on, but I really want to know what on earth is going on. It makes total sense to first power up the device and then enable it, but I don't think the heating up part should happen.
Does anyone have an explanation?
And thank you very much for everyone that showed interest in the issue and sent suggestions