This one is a bit outside of my specialty, but I'll chip in with what little I know. The 9th episode of Band of Brothers is based off the liberation of the Landsberg Concentration Camp (which was, as I understand it, a satellite camp of Dachau). Gen. Mark Clark was the one who gave the order to have the civilians from town assist with the burial details. This wasn't his brain-child, however, and came out of Eisenhower and Marshall's desire to make the Nazi genocide as public and documented as possible (largely to prevent people in the future claiming that it was all a propaganda ploy by the Allies). Eisenhower and Marshall (FDR's Army Chief of Staff) were keen to get members of Congress, as well as other high ranking allied generals, in the actual camps to personally attest to the atrocities committed. Gen. Clark's insistence that the civilians near Landsberg take part in the burial detail was an extension of this theory (that as many people as possible should personally witness the horrors so as to shoot down any future attempts to claim that it had never happened).

I don't know that it was a "widespread" practice, per se, but I do know that Landsberg wasn't the only place where civilians were enlisted to help bury the dead. Christopher Browning's 'Ordinary Men' is a great resource, as is Lucy Dawidowicz's 'The War Against the Jews.' (As a side-note, Stephen Ambrose and his book 'Band of Brothers' are not considered very trusted resources. Take anything you see in the mini-series with a grain of salt).

EDIT - Grammar