There is a long article in the current New Yorker about the case of Cameron Todd Willingham. Willingham was executed in 2004 after being convicted of setting a 1991 fire which killed his three young children. The prosecution case relied on testimony by fire investigators that the fire was deliberately set. According to the article this testimony relied on theories of fire behavior that are now known to be wrong and that there was no real evidence that the fire wasn't accidental. Assuming this to be the case the remaining evidence against Willingham is not convincing beyond a reasonable doubt. This is sufficient to show Willingham was innocent in the legal (they didn't prove it) sense although not necessarily in the factual (he didn't do it) sense. However it does seem quite possible he was factually innocent as well and just very unlucky.
If Willingham was factually innocent this was an individual tragedy but it it is not clear to me that it has any wider implications. There have been over 1000 executions in the United States since they resumed in 1976. According to Blackstone "better that ten guilty persons escape than that one innocent suffer". Still this standard does not require the exclusion of all doubt. And in practice it seems apparent that a few percent of convictions are in error. Allowing for a higher standard in death penalty cases a 1% false conviction rate seems a reasonable estimate to me. So this would give an over/under of about 10 for the number of executions of innocents since 1976. Perhaps this is high but I don't think it is at all reasonable to have confidently expected everyone executed was factually guilty. So I don't really see why this case should change any one's mind about the death penalty, any honest supporter should already have conceded that it will result in the occasional execution of an innocent.
Mistakes are made all the time that result in the death of innocents. Vaccines occasionally kill. Simple operations go wrong. Planes collide. Cars crash. So I don't see why the criminal justice system should be expected to avoid all fatal mistakes. Which isn't to say of course that reasonable efforts shouldn't be taken to minimize such errors.