Let $r(z)$ be a monic rational function (that is, a ratio of monic polynomials) all of whose finite zeros and poles lie in the unit disk of degree $n$.
QUESTION: Is there a nice upper bound on the residues of $r$ at its poles, perhaps in terms of $n$, but independent of the locations of its zeros and poles?
No, there is no such bound independent of the location of zeros and poles. Consider for example $$ r(z) = \frac{z^n}{(z-a)(z-b)^{n-1}} $$ with $a,b \in \Bbb D \setminus \{ 0 \}$. $r$ has a simple pole at $z=a$ with residue $$ \frac{a^n}{(a-b)^{n-1}} $$ and that becomes arbitrarily large when choosing $b$ close to $a$.