Should I call professors on their office phone for inquiring about research internship opportunities?
What's an "office phone"? 

We used to have these white (sometimes black) boxes with buttons on them many years ago, but the last round of budget cuts took them away. I don't think many people noticed their disappearance. 

On a more serious note, you're sounding a bit like a stalker or an annoying salesperson. If the emails aren't being answered (you should send one follow up after the initial one, but no more than two total), take that as a strong NO**. Phoning the professor is only going to annoy them. 



** You didn't ask, but reasons for the NO: 


I have no internships.
I have internships but they've all been allocated.
I have internships but they're restricted to our undergraduates or to local high school students who are in the internship program.
A quick scan of whatever you sent me makes it clear you're inappropriate (you can't spell, you just graduated high school, you have no background in the field, our internships are restricted to particular categories of enrolled students, etc.).  However, HR told us to never say 'no' in terms of justifications that could open us up to lawsuits, so it's best to just ignore. 
Internships are handled through a committee or other open process that if you did some googling you could easily find.  And I really have no use for an intern who can't find things by themselves.


As to why I don't send a 'no': I have five grant proposals, an external tenure review, two journal reviews, and my own research papers to write. Writing a polite no takes more time than a simple 'no' and again opens me / my university up to a lawsuit. Finally, answering what is essentially spam e-mail is my lowest priority.

If you are going to phone anyone, phone the department front office, university HR, or university student services or career services program (assuming you're an enrolled student). Be prepared for a 'no' answer and don't badger.