How do I fix a sticking toilet refill float?
Try to clean the valve by following these steps:


Shutoff the water to the toilet.
Flush the toilet to empty the tank (doesn't have to be completely empty, but you want some working space).
Open the valve by lifting the float to the top position, pressing down on the black plastic cap that covers the valve, and giving it a 1/8 to 1/4 turn. You should then be able to lift the cap directly off.
Clean sediment or other buildup off any small components of the valve, and if anything is easily detached, remove it temporarily for the next step. Be sure to check inside of the cap where the lever is attached.
While the cap is off, place a cup over the top of the valve and open the water supply for a few short bursts. The cup prevents a small geyser from appearing in your bathroom and the water will drain into the toilet tank (this is why you flushed it earlier). This process will clear any sediment out of the valve and supply line, including any sediment that may break free from the toilet shutoff valve.
With the water shutoff again, reassemble the valve, and attach it in the reverse process (lifting the lever on the float allows you to turn the cap with a little downward pressure on the cap).
Cross your fingers and turn the water supply back on.