Yes, bioassemblers in normal mode (there is a easier setting) will die while idling and die 50% slower when they are working, you need to constantly feed them with nutrients to keep them alive
This is incredibly misleading. You don't have to constantly feed them with nutrients to keep them alive. Once they run out of HP they transform into a different building that has a recipe in it to "revive" the bioassembler, which then has full HP again. So it's not that you're keeping them alive, it's that you're repeatedly reviving them. The building is constantly dying, but never allowed the sweet release of death, forced to live out this half-dead torturous existence as a leafy slave to the factory.