Gain tuning issues of op-amp
It appears that your op-amp is latching up on high gain. Consider what happens on start up. The output is at zero, the + input is at zero (held initially by the capacitor) and the - input is above zero at voltage determined by the divider R1, R2.

$$V- = \frac {5 \cdot R2}{R1 + R2}$$


input higher than + input will drive the output to negative rail (0 V).


The data sheet says


  Note 1: The amplifier inputs are connected by internal back-to-back
  clamp diodes. In order to minimize noise in the input stage, current-
  limiting resistors are not used. If differential input voltages
  exceeding ±1.0V are applied, limit input current to 20mA.


I suspect that you're dropping the value of R2 low enough to cause a problem here. Try putting a fixed value resistor in series with R2 to limit the maximum gain.

Edit:

Brian Drummond has commented on the original question:


  As you are AC coupling the input, you obviously don't need the same
  high gain at DC. A suitable capacitor in series with R1 will reduce
  the DC gain to unity, which should eliminate this latchup. You want C
  to be at least R3/R1 * C1 or at least 50 uF.


This is the key insight missing from my answer above.