Which example of energy could this be

If you were to take a rubber band, and stretch it with your thumb and pointer finger... which type of energy would your fingers be fighting against as they got tired?

I know that the state of the rubber band as you're holding it would be potential energy, correct? But as your fingers get tired, which force would they be fighting against? Would it be kinetic? since the rubber band wants to snap back into the original position?

only when the rubber band is snapping is kinetic energy released. If ther band is not in motion it is potential energy. The similar thing occurs when a bridge supports a weight. The weight is an expression of potential energy even though the bridge is straining to support it... same thing with a weight suspended by a string.
Incidentally the first finger is technically called the "index" finger.