How do you know when a cultured item is no longer safe to consume?
Cultured milk products rarely become unsafe.
Yogurt in particular is so acidic and teeming with bacteria already that it can't really go bad per se. It will get moldy as others have said.
I culture my own buttermilk, yogurt, kefir, cheese, etc. Many times they stay on the counter for days at a time. They get more and more sour (if they are the kind that incubate at room temperature) but they can't really go bad. Even sour cream doesn't become dangerous- the mold just tastes bad.
To second what kajaco said: if I do have cultured dairy get too sour or moldy I will wipe off the mold and bake something with it.
It should be noted that non-cultured milk is completely different. Of course it goes bad fast but after even a short time it is too bad to bake with. And never ever EVER use old milk to culture your own yogurt or cheese. The bad bacteria will overwhelm the good and you will end up with liquid that will haunt you in your nightmares!
-- Edit --
I did some research and it seems that they can eventually go bad. It doesn't fit my experience but I'll defer to the scientists. Trust your nose.