Having the same problem, coincidentally after the Oct "update" of Flash Player. Now suddenly YouTube audio defaults to mute and must be manually reset for each and every video played on MacBook Pro running Mountain Lion. None of the following methods fixes the problem and I'm suspecting it's yet another Google bug in YouTube.
1. Uninstalled Flash, rebooted, checked for any leftover preferences and cache files (there were none), reinstalled latest Flash Player - YouTube still defaults audio to mute.
2. Updated Flash Player settings (both through System Preferences and through the online Flash app) and applied recommended fixes from multiple web sites (clear stored data, allow 3rd party content, etc) - YouTube still defaults audio to mute.
3. Opened Audio MIDI Setup (output currently at 44.1 KHz), changed to 96 KHz, changed back to 44.1 KHz - YouTube still defaults audio to mute.
4. Opened GarageBand and played a few notes on piano instrument - YouTube still defaults audio to mute.
This is not a speaker setting. All audio works fine on the system, including non-YouTube videos. I've tried both the Flash and HTML5 options and they both have the same bug - YouTube audio defaults to mute. Hovering over the speaker icon in the video player shows it on mute. I have to click the icon, then drag the volume slider up in order to hear sound. But, the setting is not preserved. Next YouTube video has audio reset back to mute. Very ****** annoying.