The size of a star is a balance between pressure and gravity. Gravity pulls the outer layers of the star inward. Pressure pushes those layers upward. But increased density does increase gravity. So, as you add mass to a white dwarf, the gravity increases, but the pressure only changes a small amount.
Gravity wins and the star shrinks.
Notice that the mass-radius relation for white dwarfs means you cannot keep adding mass to a star, for eventually its radius goes to zero. If the mass can not be shed they will become neutron stars or black holes. Since they are reminants of stars born in the past, their numbers build up in the Galaxy over time. It is only because they are so faint that we fail to detect any except for the very closest ones. Once a white dwarfs contracts to its final size, it no longer has any nuclear fuel available to burn.