This is a combination of Moss (the thick rounded lumps) and lichen (the very thin plants) It is no worse for your health than having the same plants growing on your footpath or driveway or rocks.
However, while not a risk to you, you don't want this stuff growing on your roof, holding water, and rooting into your roof via the cracks. If the shingles were installed properly, there will be a sealing layer underneath which will stop anything coming through.
Since its a rental, its your landlord's problem, not yours.
If it were mine, I'd wait for a warm dry period of the year, and on a low/no wind day I'd get up and try one or more of:
- Scratch out the moss using an old screwdriver
- Broom/sweep off the roof in several directions
- waterblast (though the high pressures can damage roofing materials, use with caution)
- Spray with a suitable moss/lichen weedkiller
Wait a week for the spray to work, then on the next weekend:
- Sweep again to remove dead plant material
- Apply a sealant of some sort - either some roof paint suitable for your tiles, or a thicker mastic-type paint to seal up the gaps but still be flexible for temperature changes.
The idea is to stop the plants from rooting into the roofing material.
If the tiles/shingles were cracked and damaged, it would be better to just replace them, but these look to be in good condition.
Working on top of a roof generally means working at height, so do be careful.