No they can't, but it happens every day.
In the United States we enjoy civil law and Christians can legally marry anyone they wish. Non-Christians have the same civil rights.
Answer: First off there is no "have to" around religion involved in getting married except in the minds of the parties involved. A Christian and Buddhist (or any other combination of faiths) can marry or date and:
- agree never to discuss religion and each go their own way
- both attend both services
- either convert to the faith of the other
- both go to a third faith related to neither
- both become atheists
It is, however, wise to talk these things over before they become a problems. It is also wise not to let the families interfere as it is not their lives.
When it comes to the marriage ceremony the service may need to be a civil service if the clergy of the sides cannot act like adults.