Are there any example of $II_1$factor $M$ with maximal abelian von Neumann subalgebra $A$ and nonzero derivation $\delta:M\rightarrow B(H)$ such that $\delta(a)=0$ for every $a\in A$?
Let $M= L \mathbb F_2$ and $H = \ell^2 \mathbb F_2$, where $\mathbb F_2 = \langle a,b \rangle$ is the free group on two generators and $B(H)$ is a bimodule via the left and right multiplication with the leftregular representation $\lambda \colon L \mathbb F_2 \to B(\ell^2 \mathbb F_2)$. Define $\delta(x) = [x,\lambda(a)]$. Then, $\lambda(a^{\pm1})'' \subset L \mathbb F_2$ is a MASA (as can be shown) and $\delta$ vanishes on it. However, $\delta(\lambda(b)) \neq 0$ so that $\delta$ does not vanish on $L \mathbb F_2$. 


Sorry to be naive but why can't you just represent $M$ on a separable Hilbert space and take the inner derivation induced by any noncentral element in the masa? This maps into $M\subset B(H)$, vanishes on the masa, but is nontrivial on $M$ because the element is noncentral. 

