Is there any data anywhere showing if amateur radio operators are more likely to get cancer from exposure to RF?
FCC's page on this at https://www.fcc.gov/general/fcc-policy-human-exposure which links to several other documents including FCC OET Bulletin 56, "Questions and Answers about Biological Effects and Potential Hazards of Radiofrequency Electromagnetic Fields", last updated 1999.
This document very carefully explains RF radiation and how ionizing radiation (UV and higher) causes DNA damage.
It then explains the types of known effects of non-ionizing radiation (mostly heating which can result in burns) and the mechanisms involved.
It then goes into detail about how effects and damage other than heating have been reported, but that no conclusive evidence of this damage has been found.
However, it is also very careful to say that just because evidence has not been found, there is no proof that there is no damage either, and that further research is both needed and being done.
This 36 page document is full of detail and includes additional more detailed references. If you have unanswered questions on this topic, it is worth the read.
Short answer: we're not sure, but probably not.
Longer answer: RF energy at any frequency is dangerous if power levels are high enough. Different frequencies cause different damage. Part of getting an amateur radio license is learning this so you know how to operate safely, both for you and those around your equipment.
Update: FCC now requires most amateur radio stations do to an environmental study to determine if possible exposure levels are safe. See http://arrl.org/rf-exposure-calculator
See also http://arrl.org/rf-exposure