At least not in theory. While it is customary to swear on a Bible and say, "So Help Me God", it is not required, and you will not be forced to do it if you object. If you wish to swear on a different religious object, or if you wish to make it an affirmation, you can.
However the tradition which has been brought down from English Common law is quite strong. As late as 1939, no less than 5 states and The District of Columbia would not excluded testimony from atheists, and in a dozen more any such testimony was routinely attacked on grounds that without belief in God to keep you to the truth, you lacked credibility. I strongly suspect that there are courtrooms today (particularly in the Bible Belt) meeting the latter description. Certainly there are potential jurors who would be swayed by that argument.
Something to think about when you see atheists like me rooting for maintaining the separation of Church and State.
Cheers,
Ben