Can I mention unpaid internship experience on my resume and LinkedIn profile?

You can indicate any position you've held in you resume, as long as there is someone who had a supervising role for you or an HR department that can be contacted by a potential employer to confirm that you have indeed held that position and that the company exists. In other words, just make sure that a third party confirmation of the job is possible.

Unlike Joe Strazzere, I'd advise not to ostensively disclose the salary of the position. In general, nobody cares what your internship salary was. Be honest if asked, but I'd bet that nobody cares. Previous salaries are only important to determine if you would be willing to quit your current job (wouldn't be the case if I"m offering half you current rate) or to gauge if you are likely to find a new job paying much more than what I'm offering (maybe you got fired from a high paying position, accepted to be underpaid for the short term and is looking for something on par with your past salary, which I may not be offering, so even if I can give you a 10% raise on your current position, I cannot match your previous one, hence you are likely to leave the job I'm offering because you kept looking and found something better, which I should have expected).

All that being said, there are very few situations in which an "unpaid" internship or whatever position is legally acceptable in a for-profit business. So keep in mind that some people might not be willing to confirm and document that you've held a position that stood on shady legal basis, as not to create proof and paper trail or irregularities. Best way to handle this is to talk with whoever will be vouching for this previous position.