Thanks for the great question!

"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 does seem indicated by this passage that Marley died from some sort of head ailment. We can assume one of two things: either the bandage is part of the treatment (like a poultice), or the bandage is a result of the treatment (like a way to stop bleeding after surgery).

Poultices, held to the head with bandages, 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 often pulling teeth, and antibiotics hadn't been developed yet. As the roots of the teeth are so close to the brain, it is possible that a tooth infection could have spread to the brain. According to Dr. Martin S. Spiller, D.M.D:

"In the days before antibiotics and modern surgery, dental abscess was a common cause of death. Upon occasion, especially in the case of an untreated abscess of an upper front tooth, the patient can get a brain abscess which can kill him. This brain infection is called cavernous sinus thrombosis. Another killer is Ludwig's angina. This infection is caused by an abscess of a lower back tooth. The major symptom is severe swelling under the tongue, chin and neck. The swelling may become so severe that the patient can no longer breath. 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 wealthy upper classes who had access to large amounts of sugar." (http://www.doctorspiller.com/Root_Canals.htm)

Ear infection, especially infection of the bones behind ear, called mastoiditis, could be a culprit. Again, this infection could be fatal because of the closeness to the brain. According to Dr. Jennifer L Parker, "Prior to the discovery of antibiotics, acute mastoiditis was the most common complication of acute otitis media [ear infection] and often resulted in death." (http://www.bcm.edu/oto/grand/2394.html)

According to "The Home Library of Useful Knowledge," an 1883 book by R. S. Peale, acute tonsillitis, also known as quinsy, would have been treated with a poultice. (http://curriculum.enoreo.on.ca/socialstudies/pioneer-virtual/folkmed.html)

Other illnesses, such as aneurysms and tumors, would sometimes have been treated with surgery, which has been around since the late Stone Age (http://www.brain-surgery.com/history.html). This was done with little thought to what we now consider basic cleanliness, so infection would have been highly likely.

There is the possibility he may have had head trauma (such as a concussion or fracture) from an accident, and later died from complications.

But my favorite theory is that of oral cancer. The habit of smoking cigars and cigarettes was popular in Victorian England. (http://www.wramc.army.mil/education/tobaccohistory.htm) If we can assume the tightwad Marley may have indulged in the practice of smoking, especially cigars, it's possible he may have developed oral cancer, which could explain his jaw dropping after he removes the bandage.

I hope this helps your research. If you have any further questions, please feel free to let us know!