One of my players is a Runesmith, and he likes to use the Runestrike ability. The card says that he exhausts from one to all runes on a single runic item. The question is, can an exhausted rune provide its bonus (say, permanent +1 defence for the Rune of Stone or +3 dmg for 2 boons on an attack with the Rune of Might) to the character? If it can, the exhausting runes will only limit the number of uses of Runestrike per day (they recharge only on sunrise), while I thought that they stop giving their benefits to the user of the item too.

Meanwhile, another member of the party, a Profet of Doom, can exhaust his insanity cards with the Zealot and Profet of Doom career cards to gain benefits. Does an exhausted insanity card still work as an insanity (giving penalties and so on)?

I can't give you a definitive answer on runes but I can answer the zealot question.

For a zealot (or any other career with an insanity slot) the FAQ states that exhausting it temporarily returns them to sanity.

For example, a Zealot could invest creation points or advances to
acquire an Insanity card of his choice (which the Zealot can treat as
a Talent card), even if he has not acquired one during play. However,
unless the Insanity card is currently exhausted, the negative
effects of the Insanity are in play, even if it is not currently socketed
to an active Talent slot – the card functions as both a Talent and an
Insanity card, and abides by the rules and effects of both.

(emphasis mine)

Now I do not own Black Fire path but I would hazard a guess that as any other card that is exhausted is treated as a card with no text that any bonuses conferred by Runes would no longer function while the card was exhausted. That's the (hefty) price you pay for dealing out 10+ wounds to your enemies (not damage, wounds) for using Runestrike!

To exhaust a rune, you flip it face-down.
Seems to me that would mean you lose the bonus until it recharges.

Thank you for your answers!