I rent too and I ended up making a little fringe sleeve from scrap cotton... Basically a long fabric tube that the tassels slide into, then the whole sleeve folds under the rug edge and sticks to the underside with a few dots of removable hook and loop. No adhesive on the floor and it lifts out when guests come over. Looks tidy.
On cleaning days I also lay a lightweight draft stopper along exposed fringe that faces the room and pick it up after. The bot thinks it is a wall and turns. Zero damage and no permanent hardware.
