There are a number of ways fine tuners can cause buzzing on an instrument.
As pictured above, fine tuners which are fully-tightened may end up making contact with the top of the instrument. The tailpiece vibrates when the strings are played and, because the fine tuner is attached to the moving string and tailpiece, the tuner may bounce repeatedly off of the top of the instrument causing a buzzing sound.
To correct this issue, loosen the screw (making sure not to remove it from the fine tuner) and tune the string using the peg.
Fine tuners may also cause buzzing when the screw or nut (on the top side of the tailpiece) is loose, has cross-threaded, or has a small metal burr.
To correct this issue, tighten the nut. Once the nut is tightened, loosen and remove only the screw. Run the threads of the screw lightly along the edge or surface of a wax candle and rethread the screw back into the fine tuner.
Excessive Fingerboard Wear or a Poorly Adjusted Fingerboard
Another common issue that happens with regular use is the gradual lowering/wearing of the string grooves in the upper nut. With regular tuning, strings will slowly saw their way through the upper nut (which is located at the top of the fingerboard and is pictured above). Once the nut groove is low enough, the string will make contact with the fingerboard and buzzing will occur.
Strings can also be too close to the fingerboard when the bridge is too low. Bridge height can be affected by changes in humidity (in fact, many cellists have a winter bridge and a summer bridge which they swap out during these seasons), changes in the instrument due to age, a broken or loose neck, or for bridges which were too low from the start.
To correct this/these issue(s), bring the instrument into a repair shop for an upper nut adjustment/replacement or for a new bridge.