Let $M$ be a finitely generated module over a Noetherian local ring $R$, and $a_1,...,a_r$ be an $M$-regular sequence of maximal length. Then, $M/(a_1,...,a_r)M$ has finite length?
I guess it is not true. If the statement is true, by the dimension theorem, dim$M\leq r$=depth$M$. And we know depth$M\leq$dim$M$, so dim$M$=depth$M$. But not every finitely generated module over a Noetherian local ring is Cohen-Macaulay.
Indeed, this is false. For an explicit counterexample, let $(A,\mathfrak{m})$ be any Noetherian local ring of positive dimension and let $R=A[x]/((x^2)+\mathfrak{m}x)$. Then taking $M=R$, there is no nontrivial $M$-regular sequence, since every non-unit element of $R$ annihilates $x$. So, the quotient you consider is just $R$ itself, which does not have finite length since $A$ has positive dimension.