Are there any algebraic geometry theorems that were proved using combinatorics?

Haiman's study of the isospectral Hilbert scheme (the reduced fiber product of $\mathbb{C}^{2n}$ and $\mathrm{Hilb}_n(\mathbb{C}^2)$ over $\mathrm{Sym}^n \mathbb{C}^2$) features a lengthy combinatorial argument about the combinatorics of hyperplane arrangements and their coordinate rings (Section 4, on polygraphs).

Many things in algebraic geometry can be proved using a degeneration to combinatorial objects like hyperplane arrangements, monomial ideals or toric varieties.

For instance, de Fernex-Ein-Mustata proved an inequality involving certain invariants of a singularity (e.g., the Samuel multiplicity and log canonical threshold) by degenerating to a monomial ideal. For monomial ideals the inequality is a simple consequence of the arithmetic means geometric means inequality!

Another example: A generic smooth hypersurface has no automorphisms, as can be shown by degenerating to a union of hyperplanes (which is rigid for high degree!).

The equivalence of several definitions for some Donaldson-Thomas invariants was first established combinatorially via equality of certain classes of plane partitions. Similar techniques have been used to prove further results in this direction. See for example this paper by Benjamin Young, and discussion thereof in Chapter 7 of this book.