What are the long term and short term reasons for the execution of
Louis XVI in France in 1793?

Short term reasons:  it eliminated the alternative of reinstating the
king as head of state and the reestablishment of the old regime, and
also demonstrated that the king was not "by the grace of God" (the
usual phrase for a king's authority) somehow protected, thus removing
objections to the revolution that might be based on this concept.
Long term reasons:  at the time, the revolutionaries were thinking
they were eliminating the monarchy for all time and were establishing
a democratic government  - their reasoning.  But they did not, as
history proved, so I - non-historian -  question what "long term
reasons" applied in 1793.
Later reasoning was after the fact and then suppositions and justifications.

"The Indictment of Louis XVI (December 11, 1792)":

http://www.historyguide.org/intellect/louis_trial.html