If a toilet has a flush handle that extends out to one side of the tank and the handle sticks, the problem usually isn't the handle itself but the flush mechanism that it operates. The repair is seldom a complicated affair; it probably involves straightening the flush lever or adjusting the chain. Toilets with dual-flush levers don't have chains or flappers; the repair usually involves simply loosening the handle. If a dual-flush toilet has buttons on top of the tank and they stick, you may have to loosen a nut under the tank or replace the valve.
Open the tank and loosen the nut holding the lever to the tank. Typically, the nut has reverse threads and you turn it clockwise to loosen it; check the thread direction carefully before applying too much force and possibly breaking the nut. Loosening the nut slightly may unstick the lever. If not, move on to the flushing mechanism.
Push the lever down and watch the operation of the lever arm -- the long metal rod extending from the lever inside the tank. If the arm comes in contact with anything, bend it to eliminate the contact. If it's catching on the fill tube -- the rubber hose between the fill valve (ballcock) and the overflow tube near the middle of the tank -- push the tube deeper into the overflow tube to get it out of the way. The tube may have a clip to hold it; if not, install one on the overflow tube.
Adjust the chain -- if the lever is catching on it -- by unhooking it from the lever arm and pulling the hook off the chain. Move the hook a few links toward the flush valve, press it back onto the chain and hook the chain back onto the lever. The chain should have only about 1/2 inch of slack when the lever is in its resting ("up") position.
Loosen the lock nut holding the actuator cable to the flush handle, if you have a dual-flush toilet. Try the handle, and if it works, leave the lock nut in this position. If not, loosen the nut until the handle operates smoothly.
Lift the lid from the tank and turn it over.
Loosen the nuts securing the rods that contact the buttons on the flush valve in the tank. You may need pliers to make this adjustment. This may correct the problem.
Push the buttons on the flush valve with your finger. If either one sticks, turn the water to the toilet off, flush the tank and remove the flush valve canister and replace it.