My understanding is that it is going to burn up naturally unless they do something about it. Its orbit is low enough that it feels a fair amount of drag.
I'm probably visualizing this wrong but it seems as though it might have the potential to pass by ISS every now and then.
It won't have completely "finished its burn" until it's docked. But I don't think it would ever have a higher apogee than the ISS, because that's just a waste of fuel. If it's just short of docking then I'd expect it to be either in a same-shaped orbit slightly lower than the ISS (i.e. almost circular) ready for a Hohmann transfer, or in an ellipse whose apogee just touches the ISS' orbit ready for a circularization burn.
Even if the orbits do cross an accidental collision is extremely unlikely - the objects are very small compared to how fast they're travelling, their motion is predictable years ahead, and the Progress is at an altitude where its orbit will naturally decay in a few months if nothing is done.