To get from Amboise to somewhere in Burgundy you will probably need to go through Paris. That will involve either the slow train to Paris or the slow train to Saint Pierre des Corps then the TGV to Paris. As the TGV to the Burgundy region starts at Gare de Lyon it makes sense to take the slow train from Amboise direct to Paris - Gare Austerlitz to Gare de Lyon is a lot less of a schlepp across town than Montparnasse to Lyon.
The alternative would be to travel from Amboise to Saint Pierre des Corps and the catch the once a day train that heads towards Lyon
A couple of things to note: Amboise isn't in Loir -, Loire is a department just south of the Burgundy area, Amboise is in the Loire Valley west of Burgundy, and the TGV doesn't go to Amboise. The fastest train from Amboise only travels at 120 mph.