No, it doesn't. Phones communicate with cell towers using ordinary radio waves, just like the ones in a stereo or TV. They contain no radioactive elements.
Although there is current research and controversy on this issue, it is necessary to note that smartphones do have some level of radiations. The amounts released is however very low compared to other sources we come across in our day to day life like in microwaves or the x-rays. The manufacturers are expected to abide with the standards of safety and restrict the amount of radiations produced by their equipment. However, just like any other electronic gadget, one is ever safe applying care and also reducing exposure to the device through hands-free features and screen time.
Is there a real health risk from a user's smartphone radiation? Based on evidence so far available, we can't say with scientific certainty that a smartphone radiation is a meaningful health risk at typical levels of use; they do indeed emit non-ionizing radio-frequency (RF) radiation, but this does not possess the same energy as ionizing radiation (like x-rays) to damage DNA, in the same way as x-rays or gamma rays. Why? The majority of screening studies conducted, including the WHO and US National Cancer Institute studies studies, have demonstrated no persistent association between phone use and cancer, or other meaningful health problems. Though, there is some uncertainty, especially with very long exposure and/or heavy use, so recommendations for using a speakerphone or earphones exist as precautionary measures. In a practical sense: the radiation has not yet been proven to be unsafe yet, and is likely minimal in comparison with other risk factors and actions from our lifestyle (diet, exercise, smoking, and so forth).
Honestly, most of what we know right now says the radiation from your phone isn't a big health threat. The type it gives off is called non-ionizing, which basically means it doesn't have the juice to mess with your DNA the way X-rays or other stronger stuff can. The problems people actually run into are way more boring: neck pain from staring down all day, headaches from screen time, staying up too late scrolling. I've done that myself and felt awful the next morning. That said, the research is still ongoing. If it bugs you, little habits like using earbuds, speakerphone, or just not keeping the phone glued to your ear all day are easy precautions.