Can the Landmarks Preservation Commission (LPC) force me to do work on my landmarked building?

Yes and no.

Obviously, there’s little point in marking a building as a “historic landmark” if it’s so run down, the very features that make it “historic” are deteriorating.  To this end, building owners must do whatever work necessary to keep their building in good repair, and the LPC will issue violations if you fail to do so.  In the case that 

Outside of that, no, the LPC can’t order you to do work.  It’s only when you choose to do work on your building that it gets involved.  This is particularly so if the building was changed in some way before it received landmark status.  For instance, suppose the owner built a rooftop addition not keeping with the style of the rest of the building before acquiring landmark status.  If you wanted to modify this rooftop addition in some way, the LPC would require you to do some work to make the addition more similar to the rest of the building.  The LPC’s proposed changes, however, are not binding—if you decided not to do any work on the rooftop addition after all, you would no longer be required to institute the LPC’s required changes.