Op amp output in unity-gain configuration goes to 0V for small inputs
I suspect the following note from the datasheet may describe what you are seeing:  


  
  The ALD1722/ALD1722G has complementary p-channel and nchannel input differential stages connected in parallel to accomplish
  rail to rail input common mode voltage range.  This means that with the
  ranges of common mode input voltage close to the power supplies,
  one of the two differential stages is switched off internally.  To maintain
  compatibility with other operational amplifiers, this switching point has
  been selected to be about 1.5V above the negative supply voltage.
  Since offset voltage trimming on the ALD1722/ALD1722G is made
  when the input voltage is symmetrical to the supply voltages, this
  internal switching does not affect a large variety of applications such
  as an inverting amplifier or non-inverting amplifier with a gain larger
  than 2.5 (5V operation), where the common mode voltage does not
  make excursions below this switching point.  The user should however, be aware that this switching does take place if the operational
  amplifier is connected as a unity gain buffer and should make
  provision in his design to allow for input offset voltage variations
  


Note the last paragraph. Below 1.5V the offset voltage will vary due to one of the differential stages switching off.
I would try either running from a +/-2.5V supply so the input is at the centre of common mode range, or biasing the input to 2.5V.
Also make sure you are not loading the output too much, e.g. &lt; 5k or so (I assumed not)