I'm copying/pasting from the website below.
First, contact each program and see if they offer alternate interview weekends. An alternate weekend will be less glamorous (no formal dinners, less applicants, etc) but it will still give you an opportunity to meet with faculty and wow your interviewers. In the event of an unavoidable conflict, many programs will accommodate you on this. You might end up having to pay for more of your expenses (programs usually buy out hotel floors for specific weekends at discounted rates, so you may end up paying for your own), but it’s a small price to see a school you could be attending for 5+ years.
If they don’t offer an alternate weekend, ask if they allow Facetime / Skype interviews. This is less ideal, because you’ll miss out on seeing the campus and getting to meet with people face to face–but if it’s your only option it’s better than nothing. In this case you also have the opportunity to make yourself more memorable to faculty. Instead of being one of the crowd throughout a long day of meet and greets, you’ll be a separate memory in their minds. Take advantage of this.
Many graduate programs offer shared weekend options for applicants. Say for example you got an interview to both MIT and Harvard for the same weekend. These programs are willing to split your cost of travel to Boston between them. What this means for you, is that you might be able to talk the schools into sharing the cost of your travel so that you can split that weekend up among the two programs. This option works well for a lot east coast schools. For example, you could spend all day Saturday at a school in Philadelphia, then leave that evening and travel to Pittsburgh for a day of interviews on Sunday. You miss a little of each weekend, but you get the opportunity to meet with faculty, see the campus, and meet other applicants.
In the crazy event that both programs are unwilling to offer an alternate/shared weekend or remote interview, you’ll have to decide which you’d like to attend. This is a brutal decision, and you should be as objective as possible. Interview weekends can be a lot of fun. Attending them will almost certainly increase your affinity for that program. Keep that in mind so that you don’t let yourself miss an opportunity. Missing an interview on the other hand doesn’t bode well for your application either. You can still be admitted, but you have the deck stacked against you. If you’re still serious about the program, make sure you contact faculty right away to reiterate your interest and apologize for your scheduling conflict.
Lastly, if you decide not to attend an interview weekend because of an interview conflict, DO NOT tell faculty that’s the reason for your absence. Yes, we’re telling you to lie. Well, “lie” is more appropriate. Don’t volunteer more information than you need to. A “personal scheduling conflict” is more than enough information for faculty emails. You’re essentially choosing another school over them, and while that is a completely justified choice for you to make, you could still end up getting the brush off from a few professors. Write the professors you’re interested in working with and let them know that you’re still very serious about their work and the program but won’t be able to attend. If at all possible, try to ask them some specific detailed questions about their work and keep an email exchange going. Anything you can do to engage with them will help mitigate your absence. If you feel comfortable, ask individual professors if they have time to speak with you for 10-15 minutes on the phone. Piquing the interest of even one professor could be enough to push you into the acceptance pile even if you can’t attend the interview.