It is well known that the space of connections on a vector bundle $E\rightarrow X$ is an affine space modeled on $\Omega^1(X,End(E))$.
Let $Dol(E)$ denote the space of holomorphic structures $\bar{\partial}_E$ on $E$. We easily see from the Leibniz rule $$\bar{\partial}_E(fs)=\bar{\partial}f\cdot s + f\bar{\partial}_Es$$ that the difference between two holomorphic structures $\bar{\partial}^1_E-\bar{\partial}^2_E$ is a tensor that belongs to $\Omega^{0,1}(X,End(E))$. Moreover, there is a flatness condition $$\bar{\partial}_E\circ \bar{\partial}_E=0$$ which implies that $Dol(E)$ is not affine.
What can we say anyway on the tangent space of $Dol(E)$ as a subspace of $\Omega^{0,1}(X,End(E))$?