Do I need to wash my inkstone after every use?

The short answer is yes!

Old ink will spoil the “brilliance” of the fresh ink. Also, any dried particles with be gritty, and could damage the smooth grinding surface.  The ink has a lot of glue in, and it settles in the pores of the grinding stone as it dries, becoming very difficult to remove.  Do not allow the ink to dry out on your inkstone: good inkstones have a lid to help with this.

“Wash” means rinse with clear water.  The water can be cold or just warm. Just warm helps dissolve the glue in the ink. Hot or very cold water could crack a stone which is at room temperature.  There are many places in China called inkstone-washing pond, e.g. as used by the poet Su Dongpo.

Rub the ink off with your fingers, an old painting brush, soft cloth or sponge, or good quality kitchen towel.   Do not use scratchy materials or scourers, as these will damage the surface of the stone. You can let the stone dry by itself. If the ink won’t come off, leave the stone soaking in just warm water with a little washing up liquid, then drain and rinse thoroughly.

See this link.