DNA tested for what, exactly? I guess things like fragmentary remains may not be human, but a full skull is not so easy to confuse for a donkey. Ethnicity would only be useful if the saint in question had origins that would be out of place in Italy or if they had a specific ethnicity(like St Peter's remains not having a Levantine origin).
Yes well there are other things you could do with a DNA genotype than tag ethnicity or confirm it's human. Specifically related to a similarity metric between genotypes (which is how we go about arriving at an ethnicity estimate)
For example
if said saint has any known living relatives (and we are certain of that), then this confirms the veracity of the relic.
if said saint has multiple relics of various body parts, we DNA test each one and examine concordance.
of course a DNA test may QC fail, not enough DNA, too low quality, etc. But if it passes then we potentially have dead to rights a confirmation or refutation of the relic. For this reason I expect the church would be quite recalcitrant to have it tested, because there is a possible outcome that the relic is revealed to be a fake
