It is widely believed that correctness of string theory as a physical theory will not be decided in the near future. Regardless whether this will turn out to be correct or not, mathematical concepts derived from string theory may have proved theorems whose correctness (and relevance for mathematics) is undisputed.
What are (important in mathematics) mathematical theorems which wouldn‘t have been proved without the development of string theory?
Monstrous moonshine, the famous relationship between the dimensions of the irreducible representations of the Monster group and the coefficients in the Fourier expansion of the $j$-invariant. Borcherds' proof of monstrous moonshine uses the Goddard-Thorn theorem, which comes out of string theory, specifically the quantization of the bosonic string.
1) Verlinde's formula: Let $\mathrm{C}$ be a curve of genus $g\geqslant 2$. The Picard group of the Moduli space of rank $2$ bundles on $\mathrm{C}$ with trivial determinant is an infinite cyclic group, with ample generator denoted by $\mathscr{L}$. Verlinde's formula says that $$h^0(\mathscr{L}^{\otimes k})=\sum_{p=0}^{k} \mathrm{S}_{p0}^{-\chi(\mathrm{C})},$$ where the matrix $\mathrm{S}=(\mathrm{S}_{pq})_{p,q=0}^{k}$ is given by $$\mathrm{S}_{pq}=\sqrt{\frac{2}{k+2}}\sin\frac{(p+1)(q+1)}{k+2}\pi.$$ This was proven in the early 90s by several people (Bertram-Szenes, Faltings, Thaddeus,...). The corresponding result for rank $1$ bundles (in which case the right hand side of the formula is $k^g$) is classical and easy to prove. For rank $2$ bundles the Moduli space is not smooth, let alone a torus, and so the intersection theory is very hard to determine.
2) Witten's conjecture: Consider the DM compactified moduli space $\overline{\mathscr{M}}_{g,n}$ of genus $g$ curves with $n$ markings. For each $1\leqslant i\leqslant n$ there is a line bundle $\mathscr{L}_i$ on $\overline{\mathscr{M}}_{g,n}$ whose fiber over $(\mathrm{C};p_1,\ldots,p_n)$ is the cotangent space $\mathrm{T}^\vee_{\mathrm{C},p_i}$. Let $\psi_i=c_1(\mathscr{L}_i)$. For $k_1,\ldots,k_n\geqslant 0$ define $\left\langle\tau_{k_1}\cdots\tau_{k_n}\right\rangle_g$ to be $$\int_{\overline{\mathscr{M}}_{g,n}}\psi_1^{k_1}\cdots\psi_n^{k_n}$$ if $\sum_{i=1}^n k_i=\dim\overline{\mathscr{M}}_{g,n}=3g-3+n$ and $0$ otherwise. Then define $$F_g((t_i)_{i=0}^\infty)=\sum_d \left(\prod_{i=1}^\infty \frac{t_i^{d_i}}{d_i!}\right)\left\langle\tau_0^{d_0}\tau_1^{d_1}\tau_2^{d_2}\cdots\right\rangle_g,$$ the summation being over all sequences $d=(d_i)_{i=1}^\infty$ of natural numbers with finite support. Consider the generating series $$F=\sum_{g=0}^\infty F_g \lambda^{2g-2}$$ with derivatives $$\left\langle\left\langle\tau_{k_1}\cdots\tau_{k_n}\right\rangle\right\rangle=\frac{\partial}{\partial t_{k_1}}\cdots\frac{\partial}{\partial t_{k_n}}F.$$ Witten's conjecture says that for all $n\geqslant 1$ $$(2n+1)\lambda^{-2}\left\langle\left\langle\tau_n \tau_0^2\right\rangle\right\rangle=\left\langle\left\langle\tau_{n-1}\tau_0\right\rangle\right\rangle\left\langle\left\langle\tau_0^3\right\rangle\right\rangle+2\left\langle\left\langle\tau_{n-1} \tau_0^2\right\rangle\right\rangle\left\langle\left\langle\tau_0^2\right\rangle\right\rangle+\frac{1}{4}\left\langle\left\langle\tau_{n-1} \tau_0^4\right\rangle\right\rangle.$$ (For more on this see for example Harris & Morrison, Moduli of Curves, page 71.) For $n=1$ Witten's conjecture means that $U=\partial^2 F/\partial t_0^2$ satisfies the KdV equation $$3\lambda^{-2}\frac{\partial U}{\partial t_1}=3U\frac{\partial U}{\partial t_0}+\frac{1}{4}\frac{\partial^3 U}{\partial t_0^3}.$$ Witten's conjecture was first proven by Kontsevich, and again several other people have given alternative proofs (Okounkov-Pandharipande, Kazarian-Lando, Mirzakhani,...). But I find it difficult to imagine that this result would have seen the light of day without string theory.
3) Counting rational curves on quintic threefolds, see for example Pandharipande's Séminaire Bourbaki talk. Some of the history is described here.
If the question is set on the level of mentioning important "theorems" or "computations" or "results" which
wouldn‘t have been proved without the development of string theory
i think one could easily build a very-very long list.
Maybe it would be more appropriate to speak about which "theories" wouldn't have been out there (at least in their present form) had string theory been missing.
I believe that the case of mirror symmetry is a good candidate. See also Kontsevich's 1994 paper
Edit: Regarding the comment:
would it be right to say that actual proofs (as far as they exist) are not using input from physics and that the role of string theory was to formulate the relationship?
In some sense yes (at least regarding the part "...the role of string theory was to formulate the relationship"); and this may be seen in some precise sense as the the role of dualities in string theory: In general, duality -in string theory- means that two different string theory models may come down to the same "quantitative predictions" (for exampe the same set of topological invariants) for suitable choices of their parameters. The values of the parameters are often indicated by physical arguments (which may include experimental data or phenomenological arguments) and then the duality imposes conjectures of mathematical nature.
(However, this does not exclude the possibility that the actual proofs are indeed using input from physics -i think that Witten's work has pointed to that direction but i do not have some exact reference available right now).
The following diagram outlines some general scheme of the physics-mathematics interaction through dualities of string theory:
For more details, you can see the very interesting article: Mathematics and string theory, by Marcos Marino (see especially the discussion in p. 4-5).
P.S.: Maybe it would be interesting to have a look at the list included in the first answer at the following quora's question:
Are there any applicable uses of the string theory in maths? Or does it just apply for physics?
I recall that Richard Wentworth's first paper on precise constants in bosonization formula (which is part of his PhD thesis?) extensively used computational methods from bosonic string theory. I am not sure if the methods have been justified, since his second paper used completely different methods to derive the same result. Later I was informed that Jorgenson proved the same result using much more classical methods like asymptotic expansion of heat kernel, construction and estimate of the paramatrix, etc.
To me I feel the fact that path integral and $\zeta$-function regularization methods "coincide" in actual computation for topics related to Polyakov measure is not a mere coincidence. I do not really know string theory, but this observation striked me as something deep and subtle connecting physics to mathematics.