During negotiations my boss tried to make the situation sound better by stating that I was being paid more than the other interns and that the company was tight on money and a raise would have to come from his own pocket.
I'm not as upset that I didn't get the raise
I just have a great sense of mistrust since my boss lied
[Should I] Confront him?
I would strongly suggest that you don't. Confrontational business meetings with people you need to continue to work with after the meeting should only be done as a last resort because their purpose is to break the relationship in the hopes of reforging it as a better relationship.
If, as you seem to suggest, the current relationship is acceptable to you, and you need to continue to work with him, then there is no need for such a drastic measure, particularly when there are other resolutions available.
What should I do in this situation?
First, learn from it. This is a great opportunity to learn how to work with liars and those with unethical business standards. The reality is that he may not even see this as a lie, but a simple negotiation tactic. The ethical foundation you two operate on may be so different that a confrontation will be about semantics rather than the issue at hand - your salary. So figure out how to deal with him, and figure out what he knows and does that you might be able to use in your business practices that doesn't violate your principles.
Second, renegotiate. A few weeks after the move restart negotiations, and re-position yourself mentally to do so. Recognize that you have little to lose - all he can do is say no, and it's very unlikely that he'll fire you. While you don't need the money, you need to learn a few things about business - you now have more knowledge than you did last time you negotiated, and you may now be in a better position to obtain the raise, even without calling him out on his lies.
It's been a few weeks since we moved, the business is doing well, and I'm finding that the cost of living here is different than the old location. I enjoy working here and want to continue, and so I made the leap after you refused my last raise request, but I want to re-discuss the raise now that we're here, everything is settled, and I better understand how this change in location affects my situation. I'd like a raise to $xx.
Now that you've moved, he's less likely to give you a raise, as you've already shown you're willing to put up with a lot of inconveniences in order to stay with the company, but you already know he can afford it, you're still cheaper than the other interns, and as long as your work is equivalent he would be a poor business person if he shot you down. You have to be ready to leave, though. The "I enjoy working here and want to continue" is an implied "I'm going to leave if you don't meet my needs" and he's not going to respond positively if he thinks you're stuck and won't leave.
Spend some time looking at the job market and gain some confidence you could leave and get another job without too much difficulty first and you'll have the confidence needed to have that discussion without discomfort. Note that I've worded it so there's no lie - you don't need the raise, and you've only pointed out that the cost of living is different, not better or worse, but he may push you to discuss your personal finances in order to make his case that you don't need more money.
Resist the discussion of your personal finances, cost of living, etc. Just hold your ground, knowing the other interns get that much and live no differently, and haven't needed to justify their needs.
I'm not going to discuss my personal finances, I'm trying to keep this on a professional level, so please stop asking me about my rent, etc.
If he presses you could mention your desire to visit family occasionally, which involves travel - in other words you want to mention specifically the things that cost more because you've moved due to the job - but this is not necessary or useful, and you shouldn't bring him into your personal finances, but it may redirect him if he continues to press for specifics.
Perhaps he'll refuse. Go back a few months later and attempt to renegotiate again. Each time, even if you don't get the raise, you'll get experience. Furthermore, chances are good he will eventually relent the more you ask, as long as you're not obnoxious about it. The one downside is that if you ask too often, he may not give you a great recommendation, or it may be something he specifically mentions to future employers if they call him. Not the worst attribute in the world, but it would be a negative thing to have said about you.