You'll really have to be more specific than that. Firefox's implementation should be standards compliant, from what you said it's impossible to determine if Firefox has a bug, or the app is badly written and relies on non-standards-compliant behavior that only happens to work in WebKit.
If it's the latter, then no, Firefox will not do that.
It's not a problem with the API itself. In Gecko, the time between starting playback and the audio actually beginning is perceivablely greater than in Webkit browsers. It could be resolved by simply delaying start in Gecko, but that's a less than ideal solution.
If it's the latter, then no, Firefox will not do that.