Wiring residential internet router through existing smoke detector
No. The 2011 version of the National Electrical Code made this very clear.

National Electrical Code 2011
Chapter 7 Special Conditions
Article 760 Fire Alarm Systems
760.41 NPLFA Circuit Power Source Requirements
(B) Branch Circuit. The branch circuit supplying the fire
alarm equipment(s) shall supply no other loads. The location
of the branch circuit overcurrent protective device shall
be permanently identified at the fire alarm control panel.
This branch circuit shall not be supplied through ground-fault
circuit interrupters or arc-fault circuit-interrupters.

The 2008 version was a bit vague about this, but the 2011 version leaves no doubt. You'll have to find a power source somewhere else.

While this code snippet may not be applicable to smoke detectors, it's still good advice to keep smoke detectors on their own circuit (in my opinion).