Some communities compost for the consumers. I am not sure how common that is overall across this or other nations. But in every community where I have lived that collected a separate can for yard waste (here and in California), that yard waste is composted by the collection agency. Even when they did that where I lived, I still did my own composting, but we just don't do it anymore since we don't garden as much as previously.
However, in our community the cardboard tubes with lip balm would not be allowed in the yard waste container because of the waxes, oils and what not. So without actual home-site composting, it would have to end up in the regular trash. In spite of that, it would still deteriorate very quickly at the dump, and eventually turn to compost as it filtered down to the bottom of the rubbish pile. Knowing that, I would be be willing to give them a try if I ever run across one.
I'd probably have to alter the way I use lip balm tubes though. Meaning, I wouldn't feel it safe to keep in my pocket or leave in my car because the added heat would probably cause enough softening of the contents causing some infiltration and softening of the cardboard.
