What effects do exhausted cards have?
The key term here is involve. To resolve a faceoff or problem, characters must first be involved in it. An exausted card cannot be involved in a faceoff or problem (except if it is Stubborn) and this applies to resources too. A resource played on a exausted character will not take effect in the faceoff/problem if the character cannot be involved in the faceoff. Only the involved cards contribute to the resolution of a faceoff.