Are white dwarfs the ones always absorbing mass from a companion? It seems to me every time I see stellar remnant binary systems where there is a white dwarf and another star or something, it appears that the white dwarf always pulls mass from the star.

What if the white dwarf is with a neutron star? Will the white dwarf draw mass from the neutron star? What if the white dwarf was orbiting a supermassive star(not giant stage yet)?
A white dwarf only draws mass because it is denser that it's companion star. If a nearby object was denser than the white dwarf, and exerts enough gravity on the white dwarf, then it will draw mass from the white dwarf.

If the white dwarf is with a neutron star, and the neutron star is close enough to exert a gravitational pull that can distort a white dwarf, then the neutron star will draw mass from the white dwarf.

Now, you see, stars are way less dense that stellar remnants. No matter how massive, it's the density that counts. So if the white dwarf orbits any star, it will draw mass from the star.