No. A DAC will not treat input noise and sine signals differently.
Here is a plot of a multi-tone test.
In that test, a signal of 42 tones is put through a DAC - and 42 frequencies come out.
Imagine repeating that with a signal containing 100 or 1000 (or more) frequencies mixed together.
As you introduce ever more tones, the output spectra becomes more and more crowded, until it eventually
looks like white noise spectra.
Because that's what white noise is: all frequencies are present. Any distortion introduced would be indistinguishable from deliberately present signal.