Are there nontrivial (i.e. excluding concepts that can be defined only for $p>0$) statements in algebraic geometry that hold for all fields of characteristic $p$ for all prime $p$ but are known to be false in characteristic zero?

Here are two examples. The moduli space of dimension $g$ principally polarized abelian varieties $A_g$ contains complete codimension $g$ subvarieties in any positive characteristic $p$ (for instance, the locus of abelian varieties with no nontrivial $p$torsion points), but not in characteristic $0$ (by Keel and Sadun arXiv:math/0204229). The other example is also a resul of Keel (arXiv:math/9901149). It states that a nef and big line bundle $L$ on a projective variety over a field of positive characteristic is semiample if and only if its restriction to the exceptional locus (i.e. the union of subvarieties $Z$ such that $L_Z$ is not big) is semiample. This criterion is not true in characteristic $0$. 


One other example: in characteristic $p>0$ there exist non trivial embeddings of the affine line $\mathbb{A}^1$ into the affine plane $\mathbb{A}^2$ (i.e. embeddings that are not equal to the composition of $x\mapsto (x,0)$ with an automorphism of $\mathbb{A}^2$). For example, $x\mapsto (x^{p^2},x^{p^2+p}+x)$ is an embbeding because $k[x^{p^2},x^{p^2+p}+x]=k[x]$ (take $f=x^{p^2}$ and $g=x^{p^2+p}+x$, then $g^pf^{p+1}=x^p$). It is not trivial because the degree of one component does not divide the other one. In characteristic zero, there is no nontrivial embedding of $\mathbb{A}^1$ into $\mathbb{A}^2$. This is the famous AbhyankarMoh theorem (Abhyankar, S.; Moh, T. T., Embeddings of the line in the plane. J. Reine Angew. Math. 276 (1975), 148–166.) 


Perhaps this is an example of the contrapositive of a statement in char 0 that fails in all positive characteristics. The affine line has nontrivial \'etale covers over every field of positive characteristic, yet it is algebraically simply connected in characteristic $0$. 


Tamagawa has shown that a smooth curve (of genus $\neq 1$) over $\bar{\mathbb{F}}_p$ is determined by its profinite $\pi_1$ up to finite indeterminacy, that is, that the map $$\pi_1: M_{g,n}(\bar{\mathbb{F}}_p )\longrightarrow \mbox{Profinite groups up to isomorphism}$$ is finitetoone. This is clearly false over, say, $\bar{\mathbb{Q}}$. It's not correct to conclude therefore that there are just fewer curves in characteristic $p$, in some sense. The truth is that $\pi_1$ simply retains more geometric information. 


Here is a paper by Rachel Pries and Katherine Stevenson that addresses the question: http://www.math.colostate.edu/~pries/Preprints/11dgroupreportv036.pdf Full reference: Pries, R. and Stevenson, K. "A survey of Galois theory of curves in characteristic p" , Fields Institute Communications 60 , American Mathematical Society, Providence RI, (2011), 169191. I am not wellversed in this area, but I found this paper quite readable. Some of the examples in the paper (particularly the first half of the paper) have been brought up in answers already in this thread. 


The stack $\overline{\mathcal{M}}_{g,n}$ of DeligneMumford stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$. Therefore they are defined over any commutative ring and in any characteristic. Let $\pi:\mathcal{U}\rightarrow\overline{\mathcal{M}}_{g,n}$ be the universal curve, $\omega_{\pi}$ the relative dualizing sheaf and $\Sigma$ the union of the sections of $\pi$. Then $\mathcal{L}:=\pi_{*}\omega_{\pi}(\Sigma)$ is a line bundle on $\overline{\mathcal{M}}_{g,n}$. If $p:\overline{\mathcal{M}}_{g,n}\rightarrow\overline{M}_{g,n}$ is the coarse moduli space then $p_{*}\mathcal{L}$ is semiample in positive characteristic but this fails in characteristic zero. 

