DivinylsFan They get a sickness moodlet that should expire but sometimes gets stuck. You can of course cheat it away.
If you leave the tiberium in inventory, I believe nothing happens, at least not for a while. If you leave it out, the rock will turn into a spire that's much more valuable. You can also cut it with the gem cutter (Supernatural) to get a spire and some tiberium dust that will apparently also grow into a spire. I haven't done this because it's already way too overpowered—I don't want my sims earning §40k from one rock, let alone double that from cutting it.