airplane mode turns off 3 of the 5 radios found in most smart devices. cellular, wifi, and Bluetooth. GPS and NFC must be turned off manually. (Most of the time you can then turn back on individual radios as needed even with Airplane mode enabled). I believe some people are extra sensitive to electromagnetic radiation at the frequencies at which these radios broadcast. All but the NFC radio are broadcasting in the microwave range.
Check out this sweet diagram of the Electromagnetic radiation spectrum. From:
It's interesting to note that the farthest away from the cell tower you are, the stronger your phone is radiating microwaves. When your phone is searching for service, GPS, or wifi it is broadcasting at full power. A normal cell phone's radios are omnidirectional. It blasts radiation all directions. Also, when you are moving it has to use more power.
Personally, I have my phone in Airplane mode most of the time. I use Wifi whenever available, often Wifi hotspots are much closer than cell towers. That way my wifi radio is radiating less powerfully than the cellular radio would be. I use a headset, handset, or speaker phone whenever possible. I try to make sure my phone is not close to or touching myself or anybody else. Especially when airplane mode is off. You can try this little trick for automating the process of turning airplane mode on whenever you are connected to WiFi.
As far as cancer is concerned, here is a Government study indicating a link between cellular frequencies and increased risk of cancer.
Also, according to the FCC cell phone safety guidelines for testing a phone's SAR rating. They require the phone to be at least 5mm from any body surface. SAR rating is the amount the radiation penetrates into a fake human full of water. So, anyone using a cellphone actually touching their body or less than 5mm away. Is actually not using their phone according to how the FCC rates safety. Because of the way the intensity of the radiation greatly diminishes with distance. (The ratio is an inverse square) The SAR rating when touching your skin could be WAY higher than the legal limit of 1.60.
A little about SAR ratings and the deplorable state of any enforceable regulations.
A little about microwaves from:
Microwaves are a form of electromagnetic radiation with wavelengths ranging from one meter to one millimeter; with frequencies between 300 MHz (100 cm) and 300 GHz (0.1 cm). This broad definition includes both UHF and EHF(millimeter waves), and various sources use different boundaries. In all cases, microwave includes the entire SHF band (3 to 30 GHz, or 10 to 1 cm) at minimum, with RF engineering often restricting the range between 1 and 100 GHz (300 and 3 mm).
Bluetooth and Wifi are at 2.4GHZ, and more recently with WiFi 802.11n @ 5GHz.
In the US, cellular frequencies are 800MHz-2.5GHz
GPS is at 1175MHz-1575MHz
NFC is at 13.56 MHz
You can try out these cases that say they try to direct most of the radiation out the back of your phone: pongcase.com. (I have not tested these claims, and am not affiliated)