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What is the model on the unit and do you get an error code?
Many of these units have vent sensors but as you stated, there is no mention of such on this particular heater. I suspect for some reason the module is not recognizing the flame and therefore shutting back off. Have you checked all of the wiring connections to make sure a wire did not get bumped loose during the relocation?
Also make sure the flame sensor is clean and ceramic has not gotten cracked.You can clean flame sensor with emory cloth
I do not see a sensor on it that would shut it down if not vented properly If the flame is staying on the flame sensor, it appears it would stay lit.I am going to pen to other experts to see if they have more input on this.
It is very possible the unit and pilot will not stay lit without the vent because it is not getting enough fresh air.Looking at the manual, I do see that the vent is similar to a mobile home vent.One chamber brings fresh air in and other chamber takes exhaust out.