This actually happened to me. I was frozen.. All my nice clothing in the washer with my paper tube lip balm. Oh no!!
I was expecting the worst of course and after inspecting every inch of clothing, Nothing.
The tube stayed together with all the ingredients inside. Lesson learned though.
From now on, I always check my pockets before washing my clothes.
We cannot guarantee that you will be so lucky if you wash it with your clothes. Just keep in mind, It's paper, if it gets wet, it might break apart.