When advocating a system of morality based on reason and logic, many respected thinkers like Richard Dawkins and Christopher Hitchens offer us an ancient and simple moral rule called the Golden Rule: 'do unto others as you would have them do to you'.
There has been many contentions to the Golden Rule over the centuries. One is that it doesn't take circumstances into consideration, so a prisoner may invoke the Golden Rule before the judge, rightly asserting that the judge wouldn't want to be sent to jail himself – and so ignoring his position is due to previous wrongs. The Golden Rule doesn't take preferences into account either, so it may have you imposing on others how you'd like to be treated yourself (people have different tolerance to honesty, for example). But although these small flaws are interesting, they aren't very problematic and are not the issues I intend to address.
The issue isn't even if the Golden Rule itself. It's with the claim that it represents a morality based on rationality alone, and aimed to make irrational morality obsolete. The problem is that the Golden Rule presupposes a set of premises. It presupposes you and other are equals. It also presupposes you and other ought to be kept equals.
Of course, being equals sounds like a self-evident moral premise. Except it is not rational. For it to be, it should have been derived from the natural world, and as David Hume points out we cannot derive an ought from an is. It could also be derived from another rational statement, but this would have to be ultimately based on a natural law. At best you can say it leads to a diminished amount of suffering, but how could you argue that suffering is bad?
So, why is the Golden Rule sound? Because it bases itself on an "innate" morality (which isn't all that innate). It is both learned but also (and maybe mainly) the result of natural selection working on our brains, giving us an adaptive advantage by allowing us to work better as a society. That's my guess anyway. Whichever the process that endowed us with a natural sense of morality, it is binding only to us. It is not a law of nature, it is not universal and it certainly does not come from reasoning alone.
You can use reason to elaborate on the Golden Rule. You cannot claim that it follows from reason.
Edit: Fixed a small mistake (his/he's).