This article gives many examples of manifolds endowed with a structure whose group of automorphisms is finite dimensional.

https://www.ams.org/journals/tran/1964-113-01/S0002-9947-1964-0164299-4/S0002-9947-1964-0164299-4.pdf

In this article, it is shown that the group of automorphisms of an elliptic $G$-structure defined on a compact manifold is finite dimensional.

https://projecteuclid.org/download/pdf_1/euclid.jmsj/1260541200