Mobile phones output UHF and long microwave frequency radiation; roughly 1GHz. These are not ionising radiations, which begin at ultraviolet light frequencies and above (several and above. 1eV is about 250THz, or five orders of magnitude lower in energy per photon. GHz photons have negligible effect on atoms and molecules, so there is therefore negligible risk of the mutagenic effects that go with e.g. radiation from nuclear processes.
On the other hand, microwaves induce local heating effects in living tissue. This is simply dissipative losses as ions are vibrated by the electromagnetic radiation (the $\vec{J}=\sigma\,\vec{E}$ term in a linear material. One therefore cannot altogether rule out the risk of some permanent tissue damage from mobile phone use in some case. Reasons for further believing the risk is low are:
The brain is the most heavily envasculated organ so that its temperature can be kept rock solid stable by the blood that carries heat off to dissipate elsewhere: even a $41{\rm C}$ fever doesn't often lead to brain damage, even thought the brain dies at this temperature. The brain is kept at a much lower temperature than the main body in a fever.
We now have fairly good epidemiological data for the effects of their use, given that almost all of first world citizens have had them glued to their ears for at least 10 years.
Lack of a SIM card makes no difference: the phone is still trying to talk to the network and is therefore outputting EM radiation. However, you're not likely to use a mobile for very long if it lacks one!