Typically, massage oil should not make you have a break-out on your back. Most massage oils used should not cause a person to break-out. That said, however, some people are prone to break-outs and they may experience one after the application of massage oil.
One way to minimize a break-out is to shower after the massage using soap to rinse any excess oil. Never use alcohol on the area after (i.e. hand sanitizer, or alcohol wipes), because it will dry out your skin and stimulate your skin to produce more sebum, the natural oils of the skin and hair.
If you are worried about break-outs or you know you cannot get to a shower after your massage, talk to your massage therapist. Most usually have lotion that will absorb into your skin more easily.