For $H^1$ there are many results in chapter XIII of Raynaud's thesis (LNM 119).

First (XIII.3.1) it is easy to construct of $A$-torsors which have infinite order: Let $k$ be a field, $A_0$ an abelian variety over $k$ having a point $a$ of infinite order. Pick two rational points $x$ and $y$ (e.g. $0$ and $1$) on the affine line $L$, and let $X$ be obtained from $L$ by identifying $x$ and $y$. Consider the trivial $A_0$-torsor $P:=A_0\times_k L$ over $L$. Identifying $P_x$ and $P_y$ via translation by $a$ (which is an isomorphism of $A_0$-torsors) you get $Q_a\to X$ which is a torsor over $X$ under $A=X\times_k A_0$. It cannot be trivial: if $s$ were a section, it would give rise to a $k$-morphism $s':L\to A_0$, which must be constant, but this contradicts the requirement $s'(y)=s'(x)+a$. Clearly, if $n\in\mathbb{Z}$, the class $nQ_a\in H^1(X,A)$ is just $Q_{na}$ which is also nontrivial unless $n=0$.

There are even counterexamples over a normal two-dimensional base, but the construction is harder (XIII.3.2).

In general, if $c\in H^1(X,A)$, the property that $c$ is torsion is related to the representability or projectivity of the corresponding torsor: see XIII.2.3 and XIII.2.6.