What is the typical audio to video sync threshold when it becomes noticeable?
I don't know if there's a "standard", my personal opinion - "it depends".

In general you'll get away with a frame or two out of sync, although most editors will just line it up as accurately as possible if they can, unless there's a reason you can't?

I've filmed tap dancers on a stage and given the teacher the raw footage - and had them complain the audio was out of sync. On a raw recording you're looking at between 2 and 4 frames out, depending on how far the camera is back from the stage. 

I'll notice as soon as dialogue is 3 or 4 frames out. Other sounds often sooner. I'll notice when you've got a rear shot of someone's head, and the jaw movements aren't lining up with the dialogue. But I'm not a casual observer, and neither is the dance teacher - so like I said, it depends. There will definitely be people who notice as soon as you're out by even a small amount, and there are many more who won't spot it. Almost always when something on TV is out of sync and driving me nuts, no one else is even noticing it.

Any proper editing software will let you resync the audio, and even without a clapper, it's usually easy enough to figure out, and at least get it within a frame or two. There might not be a standard, but if your audio is badly out, you are better to just spend the time to fix it. If you can notice it, and/or it doesn't look right to you, then likely at least some of your audience will be having the same experience.