Firstly, this passage from a Guardian article;
Hundreds of these women wrote to Byron, often anonymously, furtively, entreating him for a sample of his handwriting, signed copies of his work, a curl of his dark auburn hair, a clandestine meeting.And secondly, this passage from a book titled The Works of George Byron, which quotes a traveller's encounter with Byron;
"On entering the inner shop he took off his feathered cocked-hat and showed a head of curly auburn hair."The Works of George Byron
She also provided this image, which shows his hair as much fairer and lighter than is normally depicted;
I think from all this we can safely say that his hair was at very least of the auburn shade.