The DAC works by switching an array of resistor between Vref and one or the other outputs.
The currents in the resistors are only correct if the outputs are held at 0V; that's what the first opamp does, it provides a "virtual earth" at the negative input by feeding whatever current needed in to the feedback resistor to hold Iout1 at 0V.
Without the opamp, you cannot use it like that.
It may work in a different config, to give voltage out; out1 to Vref, out2 to 0V and take the output from the Vref connection.
Though the analog switches will then be working at different internal voltages and may not function.
It is one way a built-up R-2R ladder can be configured - but it's single ended and relatively high impedance, needing a voltage buffer.