We establish universal approximation theorems for infinite-dimensional geometric rough paths, i.e., we show that continuous functions on the space of infinite-dimensional weakly geometric Hölder continuous rough paths can be approximated by functions that are linear in the signature of the path. The underlying topology determining continuity and compactness can be either the norm topology or the weak$^*$ topology. Whereas considerably more effort is required to obtain the universal approximation theorem with respect to the weak$^*$ topology, this setting ensures uniform approximation on norm-bounded sets. The motivation for establishing universal approximation theorems lies in the desire to approximate quantities derived from the solution of a stochastic partial differential equation. More specifically, our universal approximation theorems form the foundations of a novel approach to e.g. pricing of forward rates within the Heath--Jarrow--Morton--Musiela framework.