I would certainly NOT trust it to tape, and if you are not allowed to put any holes in the wall at all, then the only thing you might want to try is hanging the mirror using an over-the-door hook, something like this:
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You should be able to find something similar, and likely more apt at a local hardware store.
Otherwise I’d recommend just leaning it against the wall and don’t bother hanging it until you move to a permanent home.