Yes, recent DNA evidence should overturn the conviction of a past crime, because DNA evidence is available now that was not available back then. If it shows through DNA that the person convicted was likely not responsible, the person should be freed as soon as possible. DNA evidence is reliable.
If recent DNA evidence proves innocence, then it most definitely should overturn the conviction of a past crime. When something like this happens, it is clear that the justice system is broken. Not only should the conviction be overturned, the person who was serving time should be paid for his hardships and have the ability to sue the government.
DNA evidence is basically the strongest evidence you can get in the court of law. That and video footage. If DNA evidence is presented and it goes against the original verdict, then yes, the past conviction should be overturned immediately. That goes without question. Especially if the conviction was due to eyewitness accounts.
Using DNA evidence to overturn past convictions only makes sense when such evidence is always correct. Given that contamination over time can affect DNA evidence, it should be clear that overturning a conviction would be problematic. Say the DNA evidence indicates that a person was not present at a crime. You can only say, "Their DNA was not found at the scene," which is totally different from saying, "They were not present at the scene."