A bad fuel injector should usually be a reason to stop driving. Depending on the type of damage it would likely damage the piston and/or cylinder fairly quickly if you attempted to keep running it.
That would obviously depend on the failure mode. But there certainly are failure modes which could be quite damaging, and an ECU may have limited ability to determine what failure mode is occurring, and even if it has sensors that can indicate certain failure modes, it is not always clear if those can be trusted, as they there be additional failure modes that make sensors give misleading results.
So shutting it down certainly seems sensible.