No, HDR doesn't need tone mapping. HDR literally is High Dynamic Range, which is the range of intensities it can cover. Normal graphics can span 0-255 per colour, and your standard TV can resolve 0-255 intensities per channel. HDR can span 0-100,000 and more. Tone Mapping is about taking an image with intensities as varied as that and scaling the intensities to map onto a 0-255 range. So if you have a birghtly lit courtyard scene, with shadows at intensity 200, sandy floor at 8000, and sunlit white walls at intensity 100,000, you'd take those input and map them to 0, 128 and 255 outputs accordlingly.
An HDR display could both take the HDR signal and display it in the normal range, tone mapping as needed, while working in Floating Point space for higher quality, or take an HDR signal and show output intensities that more closely match the input, so you're shadows and white walls really are as dark or bright as they would be in real life (or something similar but scaled down as needed)