Can I Read My Boyfriends Text Messages On His Phones Online?

Only the phone number dialed, date & time it was sent can be seen on the provider's website online account manager.

What you are attempting to do is a violation of the Telecommunications Act of 1996. The only way it can be provided as a printout is to law enforcement agencies, a prosecutors' office or a convened Grand Jury via prescribed written request. Previously mentioned elements would have to contact the service providers legal compliance department for this to take place.

I answer this question at least every 3 days.  If you don't trust that person, it might not be worth sticking through it with that person.  Something to consider...

Sorry this might not be the answer you wanted, but it's better than going to jail for 2 years minimum & a $10,000 fine, right?

No you can't the only way to read his messages is to look at them on his actual phone.