I mean, there has probably not been any investigation of 5G networks specifically as the underlying question is really about microwave radiation (that is, centimeter-sized—this has nothing to do with whether it was produced by a microwave oven).
Essentially the present biophysics of electromagnetic radiation divides radiation into two categories: ionizing radiation, which has about the wavelength of molecules or smaller and also has a "kick" per photon with about the energy to dislodge electrons from atoms or break chemical bonds at an individual absorption level, and non-ionizing radiation which is larger and has less kick. Visible light is a convenient border between them: you can see it because the molecules in your eyes can absorb it and experience molecular changes that then get relayed into your brain, but those molecular changes are generally not so violent that they tear those molecules apart. So smaller light—UV, then X-rays, then gamma radiation—is considered dangerous and ionizing and such; bigger light—infrared, microwaves, radio—is considered much more benign. With that said, what we are able to see correlates pretty strongly with the wavelengths that are not absorbed with water, which one of my professors at Cornell called “maybe the strongest evidence physics can offer for the theory of evolution,” as it suggests that our mechanisms of light sensitivity must come from a bunch of precursors which at some point lived underwater. But the point is that there are other animals which can see into the ultraviolet (bees?) and infrared (snakes?) through other related mechanisms, so it is of course possible to find other compounds which are more robust to breakage and can handle more kick or other mechanisms which are a little more sensitive to infrared kicks.
For anything else we would need to defer to medical professionals on whether larger-scale electronic excitations do anything other than warming up the brain, or whether that warning can cause the brain to be better-functioning or worse-functioning. For example a 2017 review article states that,
Of the numerous studies performed to explore the effects of mobile communication devices on humans, only a few have shown that cell phones and brain tumors are statistically correlated. For example, people who have used mobile phones for more than 10 years have a clearly higher risk of brain tumors. Those who are accustomed to using their mobile phone ipsilaterally presented a probability that was twice that of people who don’t [5–7]. However, most studies have not supported the conclusion that cell phones cause brain tumors [8–12]. One study reported by the Interphone study group [13] showed that there was no increase in the risk of glioma or meningioma in users of mobile phones.
So there might be correlations but they might also fall short of causation. To give examples of what scientists are thinking about there, later in this article it talks about how cell phones might be negatively correlated with Alzheimer's risk: people who use cell phones more are less likely to have Alzheimer's. This could be for many reasons that are not the obvious causation that a warmer brain doesn't develop Alzheimer's. For example, the causation could be reversed in the Alzheimer's case; maybe people with Alzheimer's are less likely to be using their phones. Or the causation could come from an independent "confounding variable", like maybe a strong social life simultaneously causes less Alzheimer's and more mobile phone usage. Similarly, maybe people who have been around for 10 years with mobile phones are older or so, and older people get more brain tumors.
These sorts of epidemiological studies have to try and control for those factors and in general that is a very hard problem to solve.
