It's too bad that something this erroneous is still being discussed. Prince Harry has already had his DNA tested when he was a child, and so has Prince William. Diana had both boys tested in 1995 when Harry was 11 and William was 13 at the request of "the Senior Royals" for just this reason.
They are both Charles' sons. Diana wasn't even involved with Hewitt when Harry was born, nor was she involved with him in the months before Harry was conceived.
As a member of the Royal Court, the daily whereabouts of Diana, Princess of Wales were recorded in meticulous detail. The records are public and can be examined by anyone who wants to see them (and they're published in The Times, anyway).
At the time that Prince Harry was conceived, Prince William (who was born 21st June 1982) was about 18 months old. He was 2 years, 3 months when Harry was born. Diana and Charles were actively still "a couple" at that time, and they were still sleeping together. Prince Harry was born on 15th September 1984.
The records are perfectly clear on this matter: Diana first met (then) Major James Hewitt in the summer of 1986. As an officer in the Life Guards - the Royal bodyguards, so to speak - Hewitt's movements are pretty much a matter of record, too, and any claim that he met Diana earlier than the official record states can be regarded as fiction.
It's unfortunate that Diana chose a lover who had red hair, it's given rise to a lot of unfair speculation. Diana might have some problems, but she knew her duty. She gave Charles an heir and a spare before she indulged in a relationship with anyone else.
Consider this:
Anyone could've obtained a sample of Harry's DNA and had it tested. Even leaving out the fact that Diana had her sons tested, if Prince Harry were NOT Charles' son, it would have been revealed well before now.
Could Harry have really gone through Eton and Sandhurst without one person somewhere, somehow, getting a sample of his DNA? It's not that hard to get hold of - a discarded bandaid, a hairbrush left in the bathroom, a sweaty facecloth, saliva, etc. It would be the work of a moment to check the DNA against the Royal bloodline - Prince Phillip's DNA is on record, as he gave a sample in order to help identify the remains of the Russian Royal family.
It's a titillating theory simply because it would cause utter devastation in the Royal Family (being illegitimate would prevent Harry's ascending to the throne, altering the succession, etc etc) and the scandal would be a wonder to behold, but it isn't going to happen.