I've had a look for an answer to this; perhaps I've missed it though, and I don't have my rulebook to hand.
There are plenty of class cards that have the conditional phrase 'While this card is exhausted...' which provides some bonus. For instance, the 'Dungeoneer' card for the Treasure Hunter is an action (arrow) and exhausting the card to perform a search, and while the card is exhausted, the treasure hunter can perform one attack during his turn without using an action. Another example is the Spiritseeker's card 'Cloud of Mist' - while exhausted any attack that targets a space within three spaces of the spiritseeker must spend a surge just to hit.
My understanding is that during the refresh phase of each hero's turn, all cards are automatically refreshed. I imagine some people would prefer that some cards can be kept exhausted - there are few instances that you wouldn't want an additional attack in the case of the treasure hunter, or for monsters to spend surges just to hit any hero near the spiritseeker or the spiritseeker herself - but my understanding is that that isn't possible.
Can someone tell me if I've missed something? Can you leave powerful cards exhausted like this for as long as the hero chooses? Many thanks...