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.
To address your other question
What if the white dwarf was orbiting a supermassive star(not giant stage yet)?
An interesting fact about main sequence stars is that the more massive the star, the lower the star's surface gravity, so the more massive you make the star, the more easily a smaller companion star, not necessarily a white dwarf but any smaller and close companion star, can draw matter away from it.
Sirius for example, is a little over twice the mass of our sun, but 1.7 times our sun's diameter. That means it's surface gravity is 2/(1.7^2) or about 70% of our sun. It's counterintuitive but largely true that the more massive a star during it's main sequence, the lower it's surface gravity. It's also true that stars like our sun grow larger over time even as it loses mass, so the age of the star matters too, but generally, more massive stars have lower surface gravity.
Trappist 1, a red dwarf star is about 1/11th the mass of the Sun and a bit under 1/8th the diameter of our Sun. That gives it a surface gravity of about 6 times our Sun's surface gravity.
That's why there's a theoretical limit of how large stars can get, somewhere around 150 times the mass of our Sun. Around or a bit above that mass, the internal heat and outward pressure from fusion pushes the outer edges to the point where the outer layers of the star are hot enough and loosely held enough by gravity that they can escape the gravitational field of the star.
White dwarfs don't always draw mass from a companion, for example our Sun will one day be a white dwarf with no companion to draw mass from. But it sounds like you are wondering what controls which way the mass is transported. That is normally controlled by whichever star is trying to grow in size. The growth carries it across the "Roche lobe" and the mass is pulled to the other star. White dwarfs don't have internal evolution that makes them try to grow, so that's why they don't normally give up mass.
Another possibility, however, is when two stars in a close binary have their orbits come together and the stars merge. The merger of two white dwarfs is thought to be an important source of type Ia supernovae. So we might not necessarily count a merger as a transfer of mass, but it shares some of the same attributes.
