Will lowering voltage increase device running time?
The answer is yes, if you use a switching, or buck, regulator. 

Most buck regulators you can buy are fairly efficient, around the 90% ish. In stepping the output voltage down to 1/3rd of the input, the input current will be only around 1/3rd of the output current. The Ah available at the output of the regulator at 3v will be about 3x the Ah available at the output of the battery (2.7x for 90% efficiency).

A series resistor will not be efficient. It's current in = current out, so going from 9v to 3v will only be 33% efficient. Most linear regulators will be slightly worse, as they waste a little bit of current to ground as well.