This is something that is already (or should be, in theory) rigorously tested for everything that's put on the consumer market (from your cheapest USB charger to your iPhone).
Realistically we've been beaming our brains for decades now without a glut of brain tumors, but we might just be getting lucky, and if we don't know what to look for it could bite us later.
The unfortunate reality is that the market is flooded with noisy devices, often cheaply produces overseas, that vastly exceed legal limits (chargers and other rectifiers, plasma televisions, powerline adapters, and much more).
Enforcement is difficult due to how widespread these devices are.
In many places, the noise floor is to high that long-range shortwave radio communications all but impossible.
There’s definitely an effect. It’s in clinical trials right now.
Essentially EM fields alternating in the low to medium frequency bands (~100khz-1mhz) can disrupt cell processes by physically jiggling the polar molecules that make up portions of miotic spindles/microtubules. Among presumably other things this effect is being investigated as a cancer fighting mechanism called 'tumor treating fields'.
The carrier frequency of mobile phones is obviously far beyond the range in question, but there could be signal modulation components that alternate RF power levels in this frequency range.