Sorry, I somehow missed this response of yours from last month too. V8 is a JavaScript engine, so if that's the kind of error/exception you're looking at, then it's likely not your issue but something Illustrator is doing (or potentially someone else's plugin).
Pushing and popping context is required whenever your plugin is messaged through any other means than via Illustrator directly; which is to say, if your PluginMain() function (which hdi_core abstracts away so you don't have to deal with it) is NOT called, then you have to push context if you want to interact with Illustrator and pop context when you're done interacting with Illustrator. An example of this is when the OS itself messages your plugin, perhaps in response to e.g. some event in a custom widget you have created completely from scratch, or in response to a timer you registered directly with the OS, or response to your interacting with a third-party library that later calls your code of its own accord.
Pushing and popping context is required whenever your plugin is messaged through any other means than via Illustrator directly; which is to say, if your PluginMain() function (which hdi_core abstracts away so you don't have to deal with it) is NOT called, then you have to push context if you want to interact with Illustrator and pop context when you're done interacting with Illustrator. An example of this is when the OS itself messages your plugin, perhaps in response to e.g. some event in a custom widget you have created completely from scratch, or in response to a timer you registered directly with the OS, or response to your interacting with a third-party library that later calls your code of its own accord.