Loop groups (spaces of maps $S^1 \to G$ where $G$ is a Lie group) are nice examples of infinite dimensional manifolds, and they are important in physics and string theory. They have a rich and interesting theory, the basics of which are developed for example in the book "Loop Groups" by Pressley and Segal. See also the paper "Unitary representations of some infinite dimensional groups" by Segal. As for interesting results, there is the recent work of Freed-Hopkins-Teleman which relates the representation theory of the loop group $LG$ ("Verlinde ring") and the twisted equivariant $K$-theory of the Lie group $G$.
In general, infinite dimensional things come up often in quantum field theory. For example the fields under consideration might be sections of a vector bundle, or say spaces of maps of surfaces into a manifold. It is important to understand these spaces of fields, because we want to do "integrals" over them.