As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a holomorphic immersion into the complex plane. Another example is given by the theorem of Behnke and Stein, which says that every connected open Riemann surface is a Stein manifold. Is there any more interesting results for open Riemann surfaces?

The results on open Riemann surfaces are not "rare". They are just well forgotten. I only list a few books which deal with open Riemann surfaces: MR0114911, MR0228671, MR0159935, MR0264064, MR1973182. It is true that there are "two many" Riemann surfaces, and not too much can be said about "all of them". However, there is a highly non-trivial classification, and some subclasses are very important. For example, the class of "hyperelliptic" surfaces of infinite genus plays a very important role in the study of Schrodinger operators and their finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838, and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class of open Riemann surfaces that was studied a lot: MR0740581.

For example, a theorem of Grauert and Röhrl asserts that every holomorphic vector bundle on a non-compact Riemann surface is trivial.

You can find this result (and its proof) in the book of O. Forster, Lectures on Riemann Surfaces.