When to clean
Cleaning a puzzle out of the box is always a good idea. Most puzzles come pre-lubricated with a very thin lubricant that is generally of low quality. Applying lubricant to a puzzle that already contains an unknown lubricant can lead to poorer performance than using purely your lubricant of choice. Also, the screws and springs of your puzzle can be covered in lubricant as well as leftover grime from the manufacturing process so it is best to clean those parts straight out of the box as well.
Another time you may want to consider cleaning your puzzle is if the performance of your puzzle has worsened significantly since your last cleaning. This could be due to a build up of plastic dust due to excessive solving or other material such as hair or dirt from repetitive dropping of your puzzle.
How to clean
The simplest and easiest cleaning method is to simply wipe the surfaces of all the pieces of the puzzle with tissues, making sure you get into the nooks and crannies each piece, even disassembling individual cubies to wipe away excess lubricant that has seeped into the piece.
Another method of cleaning puzzles is by the use of a dishwasher. The puzzle should be completely disassembled before being placed into the dishwasher in a secure location that will prevent any pieces from being washed away. Metal components such as screws and springs should not be placed in the dishwasher as they may rust. One should also be hesitant about placing the core of the puzzle in the dishwasher as water may become trapped in the threads of screw holes and cause rust in the future.
Consideration should be taken when cleaning puzzles in a dishwasher. Some puzzles may have delicate pieces that may break under high water pressure or heat stress. Some stickers may wash off depending on the glue they have been applied with. Leaving the pieces to dry over a long period of time may cause stickers to peel off, so hand drying with a towel is recommended. If in doubt, it never hurts to ask on the forums for advice before washing your puzzle in the dishwasher.
The most thorough method of cleaning a puzzle is by hand with soapy water. Once again, screws, springs and the core of your puzzle should not be come contact with water but should instead be cleaned by hand with tissues or a dry cloth.