Nowadays, most corpses are embalmed before burial. Back in Victorian times when Dickens was writing, this was not so -- as muscles would relax and then firm under rigor mortis, corpses would be "frozen" in the position in which they died. Many times this meant the jaw was slack and the mouth and eyes were open. To avoid having to look at the dead's final facial expression, immediately after death, common practice was to shut the jaw and keep it closed by wrapping a bandage around the head. Dickens' readers would have been familiar with the practice, but a much later English work additionally suggests how eyelids were kept closed: "Now my advice for those who die / declare the pennies on your eyes...."
Though Scrooge looked the phantom through and through, and saw it standing before him; though he felt the chilling influence of its death-cold eyes; and marked the very texture of the folded kerchief bound about its head and chin, which wrapper he had not observed before; he was still incredulous, and fought against his senses. But how much greater was his horror, when the phantom taking off the bandage round its head, as if it were too warm to wear in-doors, its lower jaw dropped down upon its breast!
It seems Marley died from some sort of head ailment. Either the bandage is part of the treatment (like a poultice), or the result of the treatment (to stop bleeding after surgery). Poultices were commonly used to ease pain or swelling, such as could be caused by a tooth or ear infection. When Dickens wrote "A Christmas Carol," the treatment for toothaches was pulling the tooth out. As the roots of the teeth are so close to the brain, it is possible that a tooth infection could spread to the brain. Before modern surgery, dental abscess was a common cause of death, especially in the case of an untreated abscess of an upper front tooth, the patient can get a fatal brain abscess. During and after the renaissance, but before the advent of modern dentistry, this infection was one of the most frequent causes of death, particularly among the upper classes who had access to large amounts of sugar.
I think the bandage represents the manner of his death.