are there any dangerous/inedible compounds in tomato stems?
I just looked around online a bit and it appears that they are non-toxic to humans but they are to cats and dogs - I would do bit more research though to be sure. (one site mentions using them to flavor tomato sauce)
I don't know for sure about straight up eating them, but I do know that it is safe to include them simmering in soups or broth and then fish them out before eating. They impart nice tomato-ey flavor. But, I don't know if I would try eating the stem just because even though it might not be toxic, it would probably be awfully hard to chew and basically not digestible.