Why doesn't my tester trip the GFCI?
If you have normal power at the outlet and the outlet tester does not trip it, then there are two options:


The GFI is wired backwards. A GFI has line connections (for incoming power) and load connections (for using the GFI to protect other outlets). If the incoming power is connected to the load, you will get the behavior you describe.
The GFI is defective. This is pretty common.