Does Shuriken Flip cause the unit to exhaust?

No — Shuriken Flip moves the unit without exhausting it.

Exhaustion is only the cost for a unit's Standard Move (Rule 144.2). Moving via a spell effect is not a Standard Move, and Shuriken Flip's text doesn't instruct you to exhaust the unit, so it keeps its current ready/exhausted state (see also FAQ #3678 for the same principle). The moved unit will, however, enter exhausted if it's a fresh unit being played — but that's unrelated to the spell.