If you hear the toilet running all the time, it's possibly a misaligned or worn seal underneath the flush valve -- the canister-shaped mechanism in the center of the tank -- also the culprit in many single-flush toilets that like to run and run. Other possible issues could be debris or a broken zip tie in the fill valve, the cylindrical mechanism at the left, near the water intake.
Fixing the Seal
Turn off the water supply and flush the toilet. Remove the lid and sponge out the remaining water into the bowl.
Follow the manufacturer's instructions for removing the flush valve canister. For the Fluidmaster Duo Flush, for example, disconnect the handle actuator -- a cable housing behind the dual-action handle -- and grasp the top of the canister. Rock it away from the fill valve and lift. For other makers, you may have to twist off the top of the canister, remove the overflow tube and then loosen a U-bolt, or undo plastic locking tabs and lift the canister to reveal a red rubber seal.
Clean the black or red rubber seal at the bottom of the canister with a rag moistened with vinegar and center it on the white plastic piston. If it appears rotted or deteriorated, replace it with a new seal, pulling it under a piston lip if present. Dry the surface with a clean rag. Reattach the the flush valve and handle actuator, which should now stop leaking once you turn on the water supply.