I've always checked it out with the engine off, and on a cold engine, warm engine, somewhere in the middle, all of the above.
I've never had one go bad, so I can't say what temperature to check is best, I just know what to look for.
The FPR is located on the fuel-rail, toward the right-side of the engine as memory serves. The "fuel rail" is basically the metallic lines that run to each cylinder. If you follow that rail back to where it begins you should see the FPR.
Looks like this:
