By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally Hölder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann-Liouville process. We specialize our result for five particular counting processes. Also, we investigate Hölder continuity of the limit processes for general shot noise processes.