Why do I say that? CloudFlare does not host content at all. All content delivery networks do not host content (arguable) but only effectively _relay_ content such that delivery is faster. Once the origin server goes down, so does the content. This is an integral part of how CDNs operate. They should not be subject to DMCA take-down notices either, because they don't host content -- just enable it.
Its dangerous to the public because this significantly raises the cost in starting such a service, because now you're not just enabling faster delivery of content but suddenly you're responsible for the content itself. Handling DMCA takedowns quickly becomes fast because these large corporations will abuse the process with automation, and in turn you _must_ respond actively as if each one is 100% correct. If you want a cost-effective solution then suddenly Sony has unrestrained access to delete content from your service. This is serious.
It makes sense to have services like Youtube, image hosting sites, or file hosting sites respond to DMCA takedowns because _they are responsible for the content_ and reaching out to the uploader would mean companies need to reveal personal information of uploaders (which would quickly be abused).
But CloudFlare or other CDNs and their systems _do not have such knowledge_. They don't know who is responsible for uploading content or the ability to investigate if it's infringing on someones copyright.
If Sony doesn't like someone uploading a video to YouTube, they can file away a DMCA takedown for it. Right now Youtube has the _oppertunity_ to look into this case and fight it legally if they feel it is important. The key here is that the DMCA goes to the actual content host, Youtube. But if CloudFlare was serving Youtube and got a DMCA to takedown some video, suddenly Youtube can't fight this in court because _its not a DMCA to YouTube but to CloudFlare_.
Corporations should be required to go through proper legal systems in which the content host can respond to DMCAs and fight them if they wish to. We should be afraid of giving out giant ban-hammers more than we already have.