Sawtooth waveform using DAC
Considering the note under the drawing, I decided to omit R1 and R2
  from the circuit


R2 needs to be shorted to get this to work.

Op-amps don't work properly without feedback and that's what the line to pin 16 provides. There's probably a clue in the name of the pin if you look.



EDIT about power rails

I'm mentioning this because I think your AD8055 is powered from +5V and 0V - this will cause the problems listed below (if not using a negative rail) and I don't think the AD8055 is rated to run below a supply rail of +/-4V. Looking at your picture of the output, it appears to be bottoming out at somewhat above +500mV and rather similar to what the implied lowest output level would be on this device in the data sheet.

Just in case your supply regime is wrong, this is how it should look (different chip but same principle): -



Note the negative and positive rails on the op-amp because Iout is positive for a positive reference and for the Iout pin to be held at virtual earth, the op-amp's output must fall below 0V. Here's probably a more explicit example: -



Note that for a Vref voltage of (say) +2.5V, maximum output will be virtually -2.5V with lower values heading to 0V. At no time will the output be positive unless you use a voltage reference that is negative. This is a multiplying DAC configuration and inversion is inherent.