Is it possible for a hero to burn all their fatigue, rest, burn regained fatigue, rest, and then burn all regained fatigue again in order to increase their move distance?

The fatigue tokens are only remove at the end of the hero's turn when performing a rest action.

So no, it is not possible.

Removing fatigue tokens as the rest action is performed is a common mistake that gives heroes an important advantage over the OL. And I say that from a previous experience

The above is true, but it is possible to achieve this effect with a stamina potion search item. They recover all fatigue instantly and don't require an action with them.

But yes, rest actions use an action, but don't take effect until end of that heroes turn.

The above is true, but it is possible to achieve this effect with a stamina potion search item. They recover all fatigue instantly and don't require an action with them.

Indeed, which is one reason why it's important to remember that Search cards must be returned to the deck at the end of the quest. (Not encounter, but quest.)

This (hopefully) prevents the heroes from accumulating too many stamina potions at one time.

And if the heroes get multiple stamina potions concurrently, maybe the OL should ask to cut the search deck next time the players shuffle...

Thanks for all your help