My Kohler cistern (Kohler Code 1764A-0) overflows down the overflow pipe. The float on the inlet valve (Kohler Part 6248) seems to stick and won't turn off the water flow. If I remove the lid and lift up the float with my finger, I can stop the water flow. Should I use a lubricant on the shaft of the inlet value float? If so, what should I use?

check that the fill valve was installed correctly
adjust the float level so that when full the cistern water level is around 1/2" below the over flow pipe end
what you have now is that the inlet water is not being shut off correctly and the water is still coming in and flowing out of the overflow

Well the float bar is not turning off its set to high or problem with valve seating passing water . you will probably need to change the whole assembly.

replace the whole control unit
There may be 2 problems
the small washer that shuts off the water flow when the level is reached may be swollen and not sealing properly
the section at the bottom has cracked at a thread level
so by replacing the unit you will eliminate both problems
to remove the unit, turn off the water tap, flush the system, holding the button down until all of the water has gone
remove the inlet pipe at the bottom of the cistern
undo the retaining nut under the cistern and lift the control unit out
there will be water in the bottom of the cistern so use old towels on the floor to soak up the water that will run out

if you flush and the water continues to run into the bowl the it is the flush washer that needs repalcing
If that water shuts off and the cistern fills and then continues so that it flows down the over flow pipe into the bowl the it is the water inlet control washer that needs replacing

Sticking floats or worn float valve needles or incorrect float height, tap float bowls to see it overflow stops (sticky float), remove and clean float bowls and fuel inlet passage with carb cleaner and compressed air, check float valve for wear to the rubber tip. Reset float height by slightly bending the metal tab on the float to adjust the closing height of the valve measure from the carb bowl mating surface to the top of the float when fitted then re adjust 1mm at a time higher untill fuel doesnt over flow.

You have to twist the unit before you install it to match the height of the old ballcock. You must have made it too tall and hence the water level has to be centimeters higher than the overflow tube to shut off.

Have you tried cleaning the inlet valve? It could very well be a small piece of debris stuck in it

you will need to find out the brand name of your flush/fill valve. some valves merely twist to achieve a new fill level. others work differently. UNLESS, it is the kind with the float ball at the end of a rod. In that case, just bend the rod slightly down if the tank is overflowing.

Take the lid off the cistern tank and have a look. There is a ( usually ) round float ball in there. This is what is causing it. When the cistern fills up the ball is supposed to float up and the arm it attaches to shuts off the water, usually at the opposite end to the float.
Turn the tap on the pipe to the cistern so it is nearly off and move the arm up and down by hand. See what it is getting stuck on or what is happening.

If it is not a simple sticking problem, you may need to undo the top of the cistern valve itself ( opposite end to the ball. )
Turn the water off completely to the cistern to do this or you will get squirted. :)
They are fiddly little valves and all a bit different but if you take your time and make sure you have a good light you will be able to work out what has been happening.
Hope this is some help.

Most modern flush units have built in overflow protection, if the water level rises above the flush device then water runs into the toilet itself rather than out of the overflow pipe to the outside as in older toilets.

You could always check by taking the top off the cistern and push down the float device with your finger to see what happens!

hope this helps