Since when does blood not yield DNA?
Source: Where do DNA samples come from?
Excerpt:
"Red blood cells do not have any DNA, as they lose their nuclei (the compartment in a cell that contains the DNA) as they mature. So the DNA in your blood is in your white blood cells. To get at it, scientists first spin a small sample of your blood at high speed, to separate the cells from the blood fluid. Next, they release the DNA from the cells using a detergent and a special enzyme. Finally, they add alcohol, which makes your DNA appear as sticky blob in the mixture."I do not know that the stains on the shroud would have DNA, but dried skin cells ought to. Just thinking from a layman's viewpoint.