True or False: The Austro-Prussian War in 1866 resulted from Austria's opposition to unification.

I would say that this statement is false.  Austria was not opposed to the idea of Germany uniting.  What Austria opposed was the idea of Germany uniting under Prussian domination.  Austria wanted Germany to unite, but it wanted to dominate the new, united country of Germany.

At this point in history, the area that is now Germany was divided into many little states.  Prussia was the biggest of these and the most powerful.  It wanted to unite Germany and it wanted to dominate the new country.  However, Austria was also an ethnically German country.  It wanted to create a larger united Germany than Prussia wanted to create.  Austria wanted a larger Germany that would include Austria and it wanted to dominate this new country.

The Austro-Prussian War of 1866 came about because of this dispute.  The two countries fought over whether a united Germany would be bigger or smaller and whether Austria or Prussia would dominate the new country.  This means that the war was about the unification of Germany, but it happened because Austria and Prussia could not agree on how the unification should happen, not because Austria opposed unification completely.  Therefore, the statement in your question is false.

Was the Austro-Prussian War in 1866 due to Austria's opposition to unification?

This is false, as German unification was not complete after the war.  The war (sometimes referred to as the Seven-Weeks War) was started over a dispute over the ownership of the duchy of Schleswig-Holstein.  Denmark occupied Schleswig, and Germany occupied Holstein.  In 1846, Denmark occupied both duchies.  The combined forces of Austria and Germany went to war with Denmark in 1864 and drove out the Danish.  Schleswig was to be ruled by Germany, and Austria was to be ruled by Holstein.  In 1864, Kaiser William I of Germany and his Prime Minister Otto von Bismark thought that Austria was becoming too powerful, so Bismark, insisted on a war with Austria in order to check its growing power in Europe.  Germany (dominated by Prussia) defeated a conglomeration of Austria, Italy, and smaller southern German states.