Does a fluorescent lamp that does not light up still draw power?
You can remove the lamps that are not working. This will only let the one good light to work. Perhaps one light was faulty already before you moved in and the other was ready to go bad. It will use less power and will be of no concern, I believe you operate it with only one lamp.

So you will have an idea what the condition of the lamps are, and using the picture you offered, these type of lamps when they are going bad or have gone bad, get dark spots on the ends close to where they connect to the fixture. The darkening occurs in the glass, not the porcelain. Take for example the last lamp on the right, where it connects has some darkening at the base, where the other 2 lamps do not. The lamp on the right will not work anymore in time, or perhaps is already not lighting up (glowing). These words are interchangeable here. If your lamps in the fixture of your room have dark spots at the base, close to the end or ends that plug into the fixture, then it is most likely that the lamps are bad. It does not hurt to leave them in that I know of, they may still draw some power.
If you prefer not to have the room heavily lit, you will do fine by removing them temporarily and replace them in 9 months before you leave the rental, or replace them with new lamps if you wish, as a good gesture towards your landlord.