Does the StageFright vulnerability affect Facebook Messenger? What can I do about it? I disabled MMS in Messaging on my phone. Is FB Messenger vulnerable?
If so, what can I do to minimize the risk?
(I couldn't find anything about MMS in FB Messenger)
libStageFright is a general media library used widely on Android.  Facebook Messenger is likely to interact with it for media, and thus you are vulnerable through it.

Your best bet is to update &mdash; to a custom ROM if the manufacturer/carrier hasn't made a patch available.  Otherwise, see Stagefright security issue: what can a regular user do to mitigate the issue without a patch?

See also: http://www.androidcentral.com/stagefright