No, FLIR doesn't see through fog and cloud.
The high moisture content of the foggy air acts like a blanket and attenuates thermal energy as well as visible light so everything looks grey to the naked eye and appears to have the same thermal signature to the FLIR.
Without thermal contrast (different temperatures of different surfaces) the FLIR can see nothing.
The thermal signature of any body is the sum of a. what the body is radiating, b. what is reflected by the body and c. what is transmitted through the body from other sources.
You can't see through double glazed windows with FLIR because nearly all the thermal energy from inside the room is reflected back into it and so doesn't escape to the outside world.