This is an interesting question. To add to what has been said about inner ear/outer ear/etc.:
There are three parts of the ear that are relevant to this discussion: the outer ear, the middle ear, and the inner ear.
The outer ear: the types of infections that occur here are the ones that would seem the most likely from headphone use. Infections here are most commonly "swimmer's ear," an infection of the ear OUTSIDE the tympanic membrane (ear drum). The ear canal gets red and irritated as the result of cerumen and moisture getting trapped in the ear. This could happen with headphones, but I'd bet it's not very common unless you are using canalphones and not keeping your ears clean or have exceptionally small ear canals.
The middle ear: the infections here are called "Otitis Media" and occur typically in children. These infections occur BEHIND the tympanic membrane. Pus and serous fluid build up behind the eardrum causing pressure and pain. These are mostly viral, but some are bacterial. They can occur as an extension of a throat infection. Sometimes, if children have lots of these, they get tubes inserted in their ears to drain them. I would have a hard time believing that these are caused by headphones to any great degree.
The inner ear: these infections are described in numerous ways, but most commonly, they are called "vestibular infections." These are inside the ear, past the oval window, in the actual part of the ear where the aural signal is translated into nerve impulses. Also, the semicircular canals are there (which give you position sense, tell you whether you are accelerating when moving, and so on). Infections in that part of the ear make people dizzy.
here's a decent picture of the ear for reference.
http://sound.eti.pg.gda.pl/SRS/Image...coust/Fig1.jpg