Do CNNs benefit from HDR images?
I would say it depends on your images themselves and the task. So my short answer would be: for your task at hand, would this help a human being to perform the task? A similar answer may apply to your algorithm.

To be more specific: as far as I understand HDR compared to traditional images, the aim is to increase the dynamic range of the pictures to make it more similar to what the human eye sees. This means that for pictures with complex lighting conditions (strong backlighting for instance), there will be a real difference between classic / HDR pictures, so that if you aim at detecting objects in theses conditions, HDR will help.

On the other hand, if you have standard lighting conditions, I believe @mikalai answer fully applies and there will be not much difference.

I know this answer relates more to the "photography" side of the question rather than the machine learning side, but I think it is important to consider both in your case.

Does this help?