It is unclear exactly what is meant by a "combinatorial identity." Forinstance, let $X_n$ denote the set of alternating permutations in thesymmetric group $\mathfrak{S}_n$ that are also cycles of length$n$. Then $$ \#X_n = \frac 1n\sum_{d|n}\mu(d)(-1)^{(d-1)/2}E_{n/d}, $$where $E_{n/d}$ is an Euler number. Assuming that this counts as acombinatorial identity, then there are many further results of thisnature at http://math.mit.edu/~rstan/papers/altenum.pdf. The onlyknown proofs of most of them (including the one above, even when $n$is prime) use the representation theory of the symmetric group.
Another identity which has no combinatorial proof is $$ \frac{1}{n!}\sum_{u,v\in\mathfrak{S}_n} q^{\kappa(uvu^{-1}v^{-1})} = \sum_{\lambda\vdash n}\prod_{t\in\lambda} (q+c(t)), $$where $\kappa(w)$ is the number of cycles of $w$, and $c(t)$ is thecontent of the square $t$ of (the diagram of) $\lambda$. This isExercise 7.68(e) of my book Enumerative Combinatorics, vol. 2. Manyother similar results appear near this exercise.