Tantalum capacitor as bypass for digital IC
What is the advantage of using a tantalum capacitor in this
  application? A 0805 or 1206 10μF ceramic capacitor is even smaller
  than usual tantalum capacitor. Can I use just two ceramic capacitor,
  100nF and 10μF?


There really isn't an advantage other than that was the easiest way to get 10uF when that datasheet was written. In 2017, I would go with ceramic there for a few reasons.


Tantalum caps can catch on fire if mistreated, at least the manganese dioxide variety. (great way to make an unsuspecting ME soil themselves, FYI) The modern polymer variety don't have this problem, but cost more.
Tantalum caps have higher ESR than ceramic, which makes them tend to generate noise on power rails with pulsed loads. This is fine as a bulk cap on the input of a regulator, if the regulator has decent ripple rejection. 


One thing you do need to watch out for is the decrease in effective capacitance as applied voltage increases. If you have a 6.3V cap and put 5V on it, it's possible that it's losing 80% of its effective capacitance. To mitigate this, use a cap rated for at least twice the voltage you're going to put on it. 


  One additional question: Is it the output voltage of this DAC stable
  enough for a voltage reference? Or could it require some kind of
  filter?


Wrong question. Since this part uses VDD as the voltage reference, you really need to ask if VDD is stable enough to act as a voltage reference. There isn't enough information here to answer that question.