Technically speaking, if your bed structure doesn’t rely on a headboard, of course you can sleep in a bed without one. Stylistically speaking, you need a headboard!
Bedroom headboards were originally intended to add a layer of protection from the cold between your head and the wall since homes in the past were not as well insulated as they are today. These days, headboards add interest to a bedroom and allow you to showcase your personal style.
Beyond that, headboards have the benefit of protecting your wall from wear and tear, helping to keep your pillows in place, providing back and neck support when you sit up in bed and, yes, they can still offer a layer of protection between you and the outside cold or heat - especially if you live in an older home that has poor insulation.
If you snag one of my storage headboards or beds, you have the added benefit of more space to stash your stuff. And if that isn’t enough, I also have beds and headboards with built-in lighting and USB ports, truly making a headboard an essential piece of bedroom furniture!