We were originally going to take a bus from Amsterdam to Paris then switch to a train which would take us to Dijon, then Beaune...but the bus we were going to take isn't running on Sundays so we have a set back.

Anyone have any recommendations on the best way to travel from Amsterdam to Beaune, France?

Why not take the train the whole way?

It looks like the easiest train is the Thalys to Brussels Midi, change to the TGV to Lyon, and then a Regional train to Beaune. The trip takes 7 hours, 40 minutes.

Do you plan to rent a car? It might be easier to take the train over the French border, rent a car and drive...?

According to the bus timetable, there are 5 departures from Amsterdam every Sunday.
http://booking.idbus.com/ScheduleSelect.aspx
But the bus journey from Amsterdam to Paris lasts longer than the train journey from Amsterdam up to Beaune

The train ride from Amsterdam (dp 7.44, 10.44, 12.44, 13.44, 14.44) via Paris to Beaune takes 7 - 8 hrs. on Sundays (6 1/4 hrs on weekdays). If you take the overnight train (sleeping cars)via Amsterdam dp 20.31 - Basel 6.47/7.29 it takes 13 1/2 hrs.