Capacitive touchscreen
Capacitive touch screens work by altering the AC electric field on a fine grid in the phone screen that's detected as a change in capacitance at that node.

You should be able to alter the capacitance by touching the screen with a grounded electrical conductor. A quick trial with my phone shows this works. 

Confirmed on EEVblog by this post."I work with capacitive touchscreens daily. When we need to control when a finger is applied to the screen, we use copper tape. The tape has a conductive adhesive that keeps it on the screen. Then, a short wire to a NO relay. When the relay is activated, it connects the copper tape to ground. This increases the response to the sensor and usually appears as a finger.
I've also been looking at using a series capacitor to isolate the connecting wire from the proxy finger. "