The short answer: I strongly recommend doing HDR if you go down the PBR route. I think they go hand-in-hand, and a lot of the benefit of PBR would be wasted if you're rendering in LDR gamma space.

Here's the long answer:

Some people will talk about physically based rendering and they'll only talk about the specular BRDF (usually the discussion will be about energy-conserving Blinn-Phong or microfacet BRDF models). If you're limited to that scope, then there's no technical reason why you couldn't implement such a BRDF while still rendering in LDR gamma space. Older rendering packages supported this for many years. However the way I look at physically based rendering, HDR/linear rendering is a fundamental part of it and I would absolutely recommend doing it.

The way I see it is like this: without physically based rendering, you're like a painter. You paint arbitrary colors to the screen, and that lets you directly control what colors the viewer perceives. With physically based rendering you're now a scientist instead of a painter. You're a scientist running a physical simulation of light bouncing around your world, with the hope that the simulation will produce convincing results by using real-world laws of physics. With that in mind, HDR is really important because it lets you use a realistic range of lighting intensities. For instance the sun will be millions of times more intense than a light bulb, and a dark room will be many times darker than a grassy field. Rendering in HDR will let you represent these ranges of intensities in your game in a natural way, which makes it easier to achieve more realistic results. It's also pretty much required if you want to be able to transition from a dark area to a light (or vice versa) and make it look convincing.

Being able to support a wide range of intensities pretty directly ties into the physically based BRDFs that are often mentioned when discussing PBR. A specular BRDF that is energy conserving will be capable of producing extremely bright highlights depending on the roughness (specular power), which is very different from a non-normalized BRDF where the intensity essentially stays constant as the roughness changes. If you're going to have such a huge range in the highlights, then it makes sense to be using an HDR renderer that can handle a wide range of intensities.