The motivation for this question is to find interesting comparison functors between $1$-category theory (Set-Categories) and $0$-category theory ($\{0,1\}$-categories).
Are there interesting/known/useful in the literature monoidal functors between Set and $\{0,1\}$?
Well, there just aren't many functors $\mathbf{Set}\to\{0,1\}$ at all. First of all, since $\{0,1\}$ is a poset, a functor to it is just determined by where it sends objects. Moreover, there are no maps $1\to 0$, but there if $X$ and $Y$ are sets, there is a map $X\to Y$ unless $X$ is nonempty and $Y$ is empty. So if $F:\mathbf{Set}\to\{0,1\}$ is a functor which is not constant, it must send the empty set to $0$ and all nonempty sets to $1$.
Thus there are exactly three functors $\mathbf{Set}\to\{0,1\}$: the constant functor with value $0$, the constant functor with value $1$, and the functor which sends the empty set to $0$ and all other sets to $1$. If you are considering the cartesian monoidal structures, then the constant functor with value $0$ does not admit any monoidal structure while the other two functors are strictly monoidal.
In the context of turning $\mathbf{Set}$-categories (ordinary categories) into $\{0,1\}$-categories (preorders), the third functor would be the standard one to use. It takes an ordinary category and it turns it into a preorder by saying $a\leq b$ iff there exists a morphism $a\to b$.