Want to show the a proof of the sequent $\forall x \forall y R(x,y) \Rightarrow R(y,y)$ must have a cut. For this question we are in the Gentzen calculus. I am even having trouble just finding a proof of this sequent first off. There are two results I know of that may be relate, namely: (i) if $\Sigma$ is a cut free proof with endsequent $\alpha$ , then every formula which occurs in $\Sigma$ is a subformula of some formual in $\alpha$ and (ii) if constant c, a relation symbol R or a function symbol f does no occur in he endsequent of a Cut-free proof $\Sigma$, then c, R or f does not occur at all in $\Sigma$. That being said I don't think that these help and the only other thing I can think of is induction on grade and mix rank.
I guess this response is a bit late but...
As you observe, a cut-free proof contains only subformulas of formulas of the endsequent. This means that no sequent in a cut-free proof of your sequent can have a formula common to its left and right sides. But there is no such proof.