This "answer" will basically restate the comments of Marty and Jason Starr.
Any variety covered by schemes of the form $\mathrm{Spec}(\mathbf Z[M_i])$, or any torified variety, is rational. And indeed Severi conjectured at one point that $M_g$ is rational for any $g$! But we know a lot about the Kodaira dimension of $M_g$ by work of Harris--Mumford, Farkas, Eisenbud, Verra, ... in particular we know that Severi's conjecture is maximally false. We have $\kappa(M_g) = -\infty$ for $g \leq 16$, we don't know anything for $17 \leq g \leq 21$, and $\kappa(M_g) \geq 0$ for $g \geq 22$. In fact we know that $M_{g}$ is of general type for $g = 22$ and $g \geq 24$. But if one only wants an example of a $g$ for which $M_g$ is not rational, then I think one can find examples much earlier (maybe $g \approx 6,7$?).
See http://arxiv.org/abs/0810.0702 for a survey by Farkas.
Moreover, the cohomology of a torified scheme can only contain mixed Tate motives. In particular, this implies properties like: the number of $\mathbf F_q$-points is a polynomial function of $q$. As Marty says there are no problems in genus zero; these should be honest-to-God $\mathbf F_1$ schemes. The cohomology of $M_{1,n}$ contains only mixed Tate motives when $n \leq 10$, but for $n \geq 11$ one finds motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$. This implies that the polynomiality behaviour changes drastically -- the number of $\mathbf F_q$-points is now given by Fourier coefficients of modular forms, which are far more complicated and contain lots of arithmetic information.
Bergström computed the Euler characteristic of $M_{2,n}$ in the Grothendieck group of $\ell$-adic Galois $n$ by point counting techniques. That is, he found formulas for the number of $\mathbf F_q$-points of $M_{2,n}$ by working really explicitly with normal forms for hyperelliptic curves. These formulas turned out to be polynomial in $q$ for $n \leq 7$ (and conjecturally for $n \leq 9$), just as they would be if we knew that the cohomology of $M_{2,n}$ contained only mixed Tate motives. By thinking a bit about the stratification of topological type, one concludes that this holds also for $\overline M_{2,n}$ when $n \leq 7$, which is smooth and proper over the integers. Then a theorem of van den Bogaart and Edixhoven implies that the cohomology of $\overline M_{2,n}$ is all of Tate type and with Betti numbers given by the coefficients of the polynomials.
There are similar results by Bergström for $M_{3,n}$ when $n \leq 5$ and Bergström--Tommasi for $M_4$. But the general phenomenon is that increasing either $g$ or $n$ will rapidly take you out of the world of $\mathbf F_1$-schemes, at least if this is taken to mean "commutative monoids".
However, I don't know enough $\mathbf F_1$--geometry to say what the answer is if one takes $\mathbf F_1$-schemes in the sense of Borger. The first nontrivial case to answer would be: are the motives associated to cusp forms for $\mathrm{SL}(2,\mathbf Z)$ defined over $\mathbf F_1$ in his set-up?