No strong associations between low levels of cell (mobile) phone use and cancer have been discovered thus far. However, from what I understand, the past studies have not shown this conclusively for all cases. This is not surprising - remember that there are many, many variables involved and they all have to be investigated properly before being able to authoritatively claim that there is no association at all.
This Times (UK) article gives a brief summary of some past findings and tells of a new study that will focus on long term users. The interest from an audio perspective is some evidence that prolonged use holding the cell phone to the ear (instead of using an wired ear bud or handsfree) may be associated with acoustic neuroma brain tumors.
From my own personal experience, I believe that I can objectively claim to perceive a significant difference, after approximately ten minutes of use, between having my mobile phone with the phone pressed against my ear and using it in handsfree mode away from my ear. This does not say that I am going to get cancer, but it does contradict, at least in my isolated case, the written claims that mobile phone use could not possibly affect human tissue in any perceivable manner.
This is a controversial area of research and, in my opinion, requires careful study and careful reporting to balance convenience, health, and other issues properly.