Not overnight, that's for sure
If you've ever accidentally left a bottle of juice on the counter overnight then silently put it back in the fridge the next morning, praying it's still good, you're not alone. Lots of people make the same mistake, but it begs the question: How long can juice sit out unrefrigerated before it goes bad? The technical answer is not as long as you might think, even if it's pasteurized juice (and most commercially available juice in the United States is pasteurized, meaning it's been heated and treated to destroy harmful bacteria). According to the US Food and Drug Administration, perishable foods that are supposed to be refrigerated, like juice, can only be left out at room temperature for two hours before it's considered unsafe to consume.
Some juice companies give their products a little more wiggle room, but it's really not much longer than the guidelines from the US FDA—and almost certainly not overnight. Tropicana, for instance, does not recommend "using chilled juices that have been left unrefrigerated more than three hours," and the manufacturers of Naked Juice, "do not recommend you drink juice that has spent more than several hours in un-refrigerated conditions." The same recommendations hold true for boxes and bottles of shelf-stable juice after they've been opened.
If you think these guidelines for proper storage of pasteurized juice are strict, you'll be horrified by the guidelines for refrigerating unpasteurized juice, which hasn't been treated to kill off the bacteria that might make you ill. Unpasteurized juice should be kept in the refrigerator at all times, and you should put back into a fridge as soon as you're done with it. But really, it's just good practice to keep your juice in the fridge at all times, unpasteurized or not. And though you might be able to drink that juice you left on the counter without any issues, you're definitely increasing your risk of getting sick by doing so—so don't be a slob, and just put that juice back in the fridge once you're done with it, where you know it belongs.