Extruder motor does not turn
It turns out the problem was a firmware issue. Using the gcode below (from David Lotts in the comments), I was able to determine that the extruder was running backwards. A flag in the firmware reversed it, solving the problem.

M82 ;set extruder to absolute mode 
G92 E0 ;zero the extruded length 
G1 F200 E20 ;extrude 20mm of feed stock