LDO input below voltage range
From what i'm reading, to get the output regulated voltage, Vin must be equal to or greater than the regulated output (Vr in the datasheet) + 3% + Vdropout, which at 250mA is given as maximum 350mV, typical 178mV. 

So for guaranteed output for all conditions, 250mA for 3.3V output, the input voltage should be at least 3.3 + 0.099 (3%) + 0.350 = 3.75V.

It is more likely to operate correctly from lower than this. This IC is supposed to work for batteries, especially Lithium Polymer ones, and it seems to have a built-in undervoltage lockout/enable feature requiring a minimum voltage to begin regulating on the output. This saves batteries from being discharged too much, and would sit in a shutdown/quiescent mode drawing only 1.6 microamps typically. 

The minimum required input voltage feature is what is causing it to output 0V when you drop the input so close to the output. It is not a "dumb" linear regulator, which an older 7805 style IC might show the behavior you are expecting.