Playing on any kind of teblecloth should be a crime!!!
There was an old Columbo episode where two chess champions were to meet for a big match. They end up at the same restaurant which luckily had checkerboard tablecloths and started playing chess: salt and pepper shakers were the kings, etc. Fortunately they ordered escargot so the shells made great pawns. (A French restaurant with checkerboard tablecloths was one of the many weak things about this episode.)
In fact a cloth could easily get in the way when playing, if the board doesn't lie flat and if it rucks up in awkward places during play etc.
Most conventions I go to have table cloths on the the table--but usually the tables are all splintery and beat up, that you want the cloth there.