Facebook have decided they won't support it (Flow is their thing, so that makes sense) but Microsoft has stepped up with some pretty serious support, including good docs for the process of converting a new RN project to TypeScript:
Not sure whether that'd play nicely with Expo without a bunch of extra work. It is quite a bit harder to convert an existing project than to start a new one with TypeScript.
This transformer: https://github.com/ds300/react-native-typescript-transformer
worked perfectly for me so far. You don't have to change your code at all but just add a single file to your project and the packager will do all the rest for you.
https://github.com/Microsoft/TypeScript-React-Native-Starter
Not sure whether that'd play nicely with Expo without a bunch of extra work. It is quite a bit harder to convert an existing project than to start a new one with TypeScript.