It's more likely that a too big frame will cause 1). Discomfort due to having to lean too far forward to grasp the handlebars (such as back, neck, and shoulder pain, numbness in the hands, and compression of the nerve that innervates the genitalia (e.g. numbness) 2). More likelihood of losing control of the bike due to the handlebars being too far away to steer precisely.
Believe it or not, one inch on a bike can translate into several inches difference in reach from the seat to the handlebars. For example, I just bought a woman specific bike in 53cm after trying a 54 cm men's bike first. We measured the difference in reach between the two and it was close to two inches, which is a lot!
Also, down the road when he grows a few more inches, you can put a longer stem on the bike and raise the seat up a bit, and he'll be fine. If he grows another 4 or 5, then he would really need a bigger bike overall. So, stick with the smaller frame for now and he'll be much safer and comfortable.