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I understand it is not straight forward as a yes or no answer, but maybe a better starting question might be what is an opamp before asking what does opamps do to sound quality and whether DACs need them or not.
Short answer no. With the proviso that the amp is being run inside the limits published in its data sheet. For example, Texas Instruments designers are enormously competent and can usually offer a higher performance solution than DAC implementers and their in house discrete transistor realisation. Probably at lower cost too. But some boutique DAC implementers think that they know better . . .
There is an old white paper from ess that shows how different op amps give slightly different performance on their evaluation boards (=all other things equal) If you google it you might find it.
Here: