Unipotent algebraic group action on quasi-affine (vs. affine) variety?
Since the quasi-affine case is so easily reduced to the affine case, one doesn't really get much extra mileage out of it.
After checking my papers I am pretty sure that I never used the quasi-affine case seriously. To the contrary, in some cases I had to reduce to the affine case anyway because one is actually losing information by passing to the quasi-affine case.
For example, for affine varieties it is clear that any two closed orbits can be separated by a global function $f$ (i.e., $f$ is constant $0$ on one orbit and constant $1$ on the other). So, this also holds for unipotent groups acting on quasi-affine varieties. But this cannot be deduced just from all orbits being closed. Consider, e.g., $\mathbf G_m$ acting on $X=\mathbf A^2\setminus\{0\}$ by scalar multiplication. Then all orbits are closed but no two orbits can be separated. The reason is of course that in $\mathbf A^2$ none of the orbits stay closed.
So why bother stating theorems in the quasi-affine case, at all? I think, this is mostly to alert that one is dealing with a property which is passed on to open subsets. So one is allowed to remove all kinds of undesired points from your variety like singularities, fixed points etc.
Let $X$ be quasi-affine. Then $X$ canonically embeds in the affine variety $\hat{X}=\mathrm{Spec}(k[X])$. ($*$)
In particular, any algebraic group action on $X$ canonically extends to $\hat{X}$.
If the group is unipotent, then orbits are closed in $\hat{X}$ (by the affine version of Rosenlicht's theorem), and hence orbits inside $X$ are closed in $X$ (I'm just using here injectivity of $X\to\hat{X}$ at the level of closed points, which holds because regular functions separate closed points of $X$).
So there's no serious difference between the two versions of the theorem, and hence it's hard to say anywhere that we "seriously" use the quasi-affine version. Still it's natural and convenient to state it in this generality, since there are many natural instances of non-affine quasi-affine varieties occurring in the context of algebraic group actions, such as many homogeneous spaces (the simplest one being $\mathrm{SL}_2$ modulo the upper unipotent subgroup).
As mentioned by nfdc23, there is an issue, namely that $k[X]$ can be infinitely generated. I think this can be fixed as follows: let $G$ be an algebraic group acting on $X$. Unless I miss something, $k[X]$ is an increasing union of finite-dimensional sub-$G$-modules. Passing to the $k$-algebras they generate, we see that $k[X]$ is increasing union of finitely generated $k$-subalgebras $A_n$. So we have genuine affine varieties $Y_n=\mathrm{Spec}(A_n)$ with canonical $G$-equivariant morphisms $X\to Y_n$.
Since $X$ is quasi-affine, there exist a finite number of regular functions separating the points, so $X\to Y_n$ is injective for some $n$ and the argument goes through.