Are there more Nullstellensätze?

If $k$ is a finite field with $q$ elements and $I$ an ideal of $k[x_1,\dots,x_n]$, then $\overline I=I+I_0$, where $I_0=(x_1^q-x_1,\dots,x_n^q-x_n)$. This follows immediately from Hilbert’s Nullstellensatz applied to the algebraic closure of $k$, and the observation that any ideal extending $I_0$ is a radical ideal (as it contains all polynomials of the form $f^q-f$).

On an unrelated note, a more explicit description for the case of $k$ real-closed follows from Stengle’s (Positiv- and) Nullstellensatz: $f\in\overline I$ iff $-f^{2n}\in I+\Sigma$ for some $n\in\mathbb N$, where $\Sigma$ is the set of all sums of squares of polynomials.

Pete, a Nullstellensatz-like result for finite fields is the "Combinatorial Nullstellensatz" formulated by Noga Alon, and it does imply the Chevalley-Warning theorem. Searching for CN will produce Alon's paper and several others on the first page of results.

This paper by Laksov addressed your question in detail:

D. Laksov, Radicals and Hilbert Nullstellensatz for not necessarily algebraically closed fields,. L'Enseignement Mathematique, 33, 323-338 (1987)

There seems to be more work on this, so a MathSciNet search on papers that cited the one above would help, I think.