This looks like an ok part for an ADC reference pin... will sink up to 15mA, and using a 1K source resistor yields only 2.7mA.
Are you sure it's connected correctly? Did you set the AREF pin for external? Is the FB tied directly to Vout? Is there a 1K resistor between the Vin and Vout/FB pins?
The only thing I see about it turning the output off is if the FB is lower than .6V, which it would turn the output off so that it can come up.
The way this part works is to sink current through the source resistor (fig 3, 1K tied between Vin and Vout) to drop voltage across it until the FB is at .6V. If you want a .6V regulator, you connect the Vout to the FB pin directly. If the FB voltage goes below .6V, the Vout turns off and quits sinking current so that the voltage dropped across the source resistor is less and the voltage at Vout (and FB) goes up. In the configuration of figure 3, when the Vout sinks 2.7mA, the 1k resistor will be dropping 2.7mA x 1K = 2.7V, so 3.3V - 2.7V = 0.6V.