Floyd Rose tremolo units are known to be very resilient, but to keep them functioning properly, you need to perform routine maintenance. Excess wear and friction can severely affect a tremolo's performance and should be guarded against by periodically cleaning and lubricating the tremolo unit. Because cleaning often requires removing the unit from the guitar, you should be able to perform a basic set-up of your tremolo unit before removing it for cleaning. Here's a guide on how to clean and lubricate a Floyd Rose tremolo.
Remove the tremolo unit from the guitar body. You'll need to remove the strings, body route covering and the tremolo springs. Carefully slide the tremolo away from the posts and out of the cavity. Play the unit on a clean cloth and lay it on your work surface.
Use another cloth to remove any buildup of grime or dust that you see. Lightly wet the cloth if necessary, or use a small amount of saliva to dissolve tough buildup.
Apply a small amount of light oil to a cotton swab and apply it to each string saddle, lightly coating all the moving parts and screws. You'll likely need to change the swab often to avoid leaving cotton particles behind.
Apply a small amount of oil to the top and bottom of the unit, and wipe any excess with a clean cloth.
Clean the tremolo posts and apply a small amount of graphite lubricant to the knife-edge slots.
Insert the tremolo into the guitar body and reattach the springs and strings, and replace the body route covering.
Some of this maintenance can be performed without removing the tremolo unit. If you don't want to remove the tremolo from the guitar, or don't feel comfortable doing so, simply clean and oil the top of the tremolo unit. To avoid wasting strings, schedule the cleaning around a regular string change.
Be careful not to damage the knife edges of the tremolo unit while removing it from the guitar.