Is there a closed form for $\sum_{i=1}^n \left(\frac{i}{x + i}\right)^{i y}$?

One thing is for sure: the terms in the sum converge to a constant, namely $$ \lim_{i\to\infty} \left(\frac{i}{x + i}\right)^{i y} = \frac{1}{\lim_{i\to\infty} \left[\left(1+\frac{x}{i}\right)^i\right]^y} = \frac{1}{[e^x]^y} = e^{-xy} $$ Convergence of the terms may be very slow, depending on $x$. Anyway, it means that the sum itself does not converge to anything: it is continuously increasing.