Dispensers and droppers are powered by redstone up to two blocks above them or directly diagonal one block above them. They remain constantly powered, thought they do not register as powered, and will no longer dispense blocks until the power is removed.

Photos can't do this bug justice, so here is a link to a short video describing the problem in it's entirety: https://youtu.be/JDDPc-alMhU

This bug was mentioned by MC-108 but that report only describes piston behavior with no reference to droppers/dispensers beyond the title, so this is for further reference on what may be the same issue with pistons.