Is the distance between the Sun and the Earth increasing?

I think the reasoning has an error. It assumes $v$ is constant, but instead we ought to assume the angular momentum is constant.

By dimensional analysis that leads to

$$r \propto \frac{L^2}{GM}$$

so as $M$ decreases, $r$ increases (the original post had $r \propto M$, not $r \propto 1/M$.

On the other hand, assuming a circular orbit seems dubious.

As the other commenters said, this effect is minute. A significant effect on the orbit of the moon around the earth is tidal evolution, which does actually push the moon further away.

Such a small magnitude makes this process negliable among other factors; indeed this result literally means that you don't need to care about this process until you build extreamly accurate theory of the full solar system dynamics. To the extent that is probably unreachable due to deterministic chaos.

The Sun is also losing mass due to the solar wind. Again the fraction of mass lost is very small compared to the mass of the Sun, so the effect is very small.

There are these relevant papers that I think you will find interesting:

Orbital effects of Sun's mass loss and the Earth's fate

Astrometric Solar-System Anomalies