I'm installing low voltage fire alarm wiring in a drop ceiling. Can I support it on the ceiling wire.

No, any cable must be independently supported to the building structure, you can use added support wires just like the ceiling is held up with, but no wiring can attached to the support dedicated for a suspended ceiling, UNLESS, the ceiling system is engineered and documented to be capable of supporting mechanical systems.

See Section 830.7 of the NEC