I am making intensive use of the input speech fields in a "web app" which is destined to be used by about 2 thousands users every day.
My understanding is that the speech input field, at least in Chrome (not sure about Firefox?) connects to Google speech servers.
Is there any sort of quota that can be hit, after which the service will stop working?
There is time limit for the length of each audio stream - this is a limit on Google's servers and appears to be around 60 seconds, although this is not documented.
The limit makes perfect sense - you can't expect Google to process hours of audio as that model would never scale. You should treat the service in a way similar to Siri - initiate a recognition session, feed it a short stream of audio, process those results, initiate a new session.
Unfortunately, because the web speech recognition work is still under active development, things are still very fluid and documentation is limited. I recommend digging around on the Chrome browser mailing lists and posting specific questions there. I have found the folks working on this at Google to be quite responsive to specific questions.
For a minimal demo of speech recognition and some thoughts on the issues with the current speech API, you might like to to look at this Tutorial on Web Speech Recognition