If $C$ is a smooth projective curve of genus $g \geq 3$ and $J(C)$ is the Jacobian of $C,$ then an Abel curve $C \subset J(C)$ is not algebraically equivalent to its image $-C$ under the negation automorphism, even though $C$ is homologically equivalent to $C.$  This was proved by Ceresa in the paper

https://www.jstor.org/stable/2007078

EDIT:  Ceresa also shows in this paper that if $C$ is generic and $1 \leq k \leq g-2$, the cycles $W_k$ and $-W_k$ in $J(C)$ (recall that $W_k$ is the cycle parametrizing effective line bundles of degree $k$ on $C$) are algebraically independent (although they are homologically equivalent).  This gives examples of non-injectivity for higher-dimensional cycles.