In general my antique and vintage backgammon boards are by nature pretty resilient. But always bear in mind that antique wood hates two things; extremes of temperature and extremes of humidity.
Avoid leaving your board in direct sunlight for extended periods (but of course you can still play backgammon outside!), and never place it on top of a radiator or directly beside an air-conditioning outlet. Heat and moisture cause the fibres in the wood to expand, and this can lead to distortion, and with marquetry in particular, cause 'rising'. Cold and dryness does the reverse, shrinking the fibres - but with a similar effect.
Wooden playing pieces can sometimes split if left for a long period in direct sunlight or extreme heat. In the event of damage (or loss) I can almost certainly replace the piece or pieces concerned with identical period pieces - as I have a large stock of playing pieces of all types and sizes. I also carry spare antique and modern dice and also leather dice cups. Just drop me a line at the email address below and I will find a solution for you.
Do remember to wax the external surface of the board from time to time as this will help to protect it too from moisture.
Marquetry, whether wood or leather, is particularly sensitive to water. Wax the internal and external surfaces once or twice a year with a good quality wax for wood / leather - the board will love this - and it will absorb the wax readily, thus nourishing the wood / leather naturally, as well as protecting it from moisture. If you accidentally spill liquid on your board, dry it immediately with paper towel before the moisture can penetrate the fibres of the wood or leather. Never use a hairdryer or similar on the marquetry.
Sadly, if left, water damage to a marquetry playing surface is nearly always irreversible.
Use a soft brush, or a vacuum cleaner (set to low), to clean dust out of the corners of the playing surface from time to time. Never try to clean a cloth playing surface with water or carpet shampoo or anything wet - trust me, this way lies disaster. A very slightly moistened sponge - using a gentle circular action on a small area can work to remove a specific small stain, but never attempt a large-scale cleaning operation using liquids. I can replace the baize playing surface in the event of severe staining or burning. Please do get in touch with me to discuss if this is of interest.