First of all, be very careful handling silver nitrate. Gloves, eye protection and an apron are necessities. Silver nitrate (silver caustic) was used to cauterize wounds and in surgery in the 19th century, so if you get it in your eyes, it will cause blindness. If you get it on your skin, you may not realize it until it is exposed to daylight and develops out and turns your skin black.
To remove it from a table, you might have to bleach and fix it in the same way you might bleach a print (potassium ferricyanide, potassium permanganate or potassium dichromate type bleaches), but this might also have an undesirable effect on the table, so test first in someplace that isn't obvious. Potassium permanganate is probably the least hazardous option, and is used in some darkroom tray cleaner formulas.