Does Fight or Flight exhaust your unit when moving it back to base?

No, Fight or Flight does not exhaust your unit when moving it back to base.

According to the official FAQ for Fight or Flight:

• "No, the unit does not exhaust. Only standard moves have an implicit exhaust cost. When a spell tells you to move, you only do what the spell says."

Additionally, Rule 407.3 states that the "Standard Move" inherent to units is a discretionary action that requires exhausting the unit as a cost, whereas card effects that instruct you to move a unit (like Fight or Flight) provide movement without requiring that cost unless the card text explicitly states otherwise.