Could the LA County Museum of Art fall into the LaBrea Tar pit during an earthquake?
The two geological faults that may be responsible of large earthquake in proximity of the museum are:


Hollywood fault;
Santa Monica fault (actually many branches, but they all go together as Santa Monica fault).


The museum seems to be enough far away from the fault trace (the surface expression of the fault itself), so it is unlikely that there will be a major shift in the soil. 

However, the museum may sink in the soil due to the earthquake mobilizing the soil, especially if composed of loose sand.

The phenomena responsible of the museum "falling in the soil during an earthquake" is liquefaction. The geological service of California provides you an interactive map to check if a certain address may be under liquefaction hazard https://www.americangeosciences.org/critical-issues/maps/interactive-map-california-earthquake-hazard-zones and ...

at the moment the interactive map is not accesible to me. Sorry :) 

If liquefaction is possible, either the tar may quickly move and sputter on/in the museum, or the museum may sink in and later tar may infiltrate slowly in the building.