Let $X$ be a comapct Riemann surface of genus $g$ and let $J\: : \: X\to \mathbb{C}^{g}/\Lambda$ be the Abel-Jacobi map. This map is a smooth embedding. Let $p\in X$ such that $J(p)=\Lambda$ and consider
$$
J\: : \: X\setminus p \to (\mathbb{C}-\Lambda)/\Lambda.
$$
Then $J^\*$ induces a surjection in complex de Rham cohomology. $J$ is injective and hence is defined at the level of the (ordered) configuration space
$$
\operatorname{Conf}\_{l}(J)\: : \: \operatorname{Conf}\_{l}(X\setminus p) \to \operatorname{Conf}\_{l}((\mathbb{C}-\Lambda)/\Lambda).
$$
Let $V\_{1}\subset H^{2}(\operatorname{Conf}\_{l}(X\setminus p), \mathbb{C})$ , resp. $V\_{2}\subset H^{2}(\operatorname{Conf}\_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})$ be the space generated by Massey products $(a\_{1}, \dots, a\_{n})$ between degree $1$ elements, for $n\geq 2$. Does $\operatorname{Conf}\_{l}(J)^{\*}$ induces a surjection 
$$
\operatorname{Conf}\_{l}(J)^{\*}\: : \: \mathbb{C}\oplus H^{1}(\operatorname{Conf}\_{l}((\mathbb{C}-\Lambda)/\Lambda), \mathbb{C})\oplus V\_{2}\to \mathbb{C}\oplus H^{1}(\operatorname{Conf}\_{l}(X\setminus p), \mathbb{C})\oplus V\_{1}?
$$
Notice that $\mathbb{C}$ represents the $0$-th cohomology group.



Jacobian and configuration space and massey products
The answer is yes.


First off, the map $\mathrm{Conf}\_l M \to M^l$ induces an isomorphism on $H^1$ if $M$ is an oriented manifold such that $\dim M > 2$, or such that $\dim(M)=2$ and $M$ has positive genus. See my answer to a previous question [fundamental group of configuration spaces of ordered points on open Riemann surfaces](https://mathoverflow.net/questions/263893/).


Secondly, if $X$ is a smooth projective variety, then $X \setminus \{pt\}$ is a formal topological space. This is because the mixed Hodge structure on the cohomology of $X \setminus \{pt\}$ is pure. See e.g. Dupont's paper <https://arxiv.org/abs/1505.00717>. So all the triple or higher Massey products of classes in $H^1(\mathrm{Conf}\_l (X \setminus \{pt\})) = H^1((X\setminus \{pt\})^l)$ vanish, because the Massey products are functorial and they vanish in the cohomology of $(X\setminus \{pt\})^l$. The same holds for the jacobian.


So the result follows from what you've already said about the case $l=1$ and the Kunneth formula.


