Will mentioning that I'm considering a PhD to start a company hurt my chances?
One thing you should really consider first, is what are your rights to take the research and start a company: in most universities, whatever discovery is made there legally belongs to them. You might be named on a patent but you may not be the legal owner of any of it, or you may need to license the intellectual property from the university.

Now, in some places, they actually encourage this kind of ventures, but it's framed within the university bound; that means they may even help you start a company, but they'll want royalties or something in return.

In the end, I'd recommend you look at where your career goals will be best received, in your field of choice, rather than apply all over and interview to discover they don't encourage this type of outcome. That'd probably save you some precious time.