Being described as a hero in a story does not inherently indicate that the character is a good person. Indeed, well-written characters live in shades of gray, in which there are moments of greatness along with moments times in which they make poor decisions as well. The Harry Potter stories are no exception.
Formed to be a secret society dedicated to battle back the forces of Voldemort, the Order of the Phoenix was a tight-knit group that was inevitably undone by the betrayal of one of their own. However, to suggest that this is the only fallacy within the organization would simply be inaccurate.
In total there were 33 total members of the Order written in J.K. Rowling's books. This list, however, will include 21 of them. The reasons for this is that, primarily, the names listed on this list simply existed to bolster the number of atrocities Voldemort and his Death Eaters committed. Yes, the Longbottoms were tortured into insanity by Bellatrix Lestrange, and people like Marlene McKinnon and Hestia Jones were killed in action, but those are virtually the only facts known about the characters.
One notable exception is Nymphadora Tonks. The shapeshifter was largely an innocent, whose greatest fault was slipping into her own depression in the sixth novel over a forbidden love with Remus Lupin. It was determined pining over a werewolf wasn't necessarily a bad thing.
With that in mind, let's examine some of the Order's more substantial shortcomings.