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 over $X$ which have infinite order: Let $k$ be a field, $A$ A_0$an abeliabelian variety over$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. 1 For$H^1$there are many results in chapter XIII of Raynaud's thesis (LNM 119). First, it is easy to construct of$A$-torsors over$X$which have infinite order: Let$k$be a field,$A\$ an abeli