Question: I understand that if $A$ is unitary, then $A^{-1} = A^\ast$ are also unitary matrices.

But if $A$ is a unitary upper triangular matrix, then how are $A^{-1}$ and $A^\ast$ also upper triangular? This was from a particular class.

Then the statement goes on to say that $A$ will be a diagonal matrix with diagonal entries having absolute value $1$.

Answer: 
If $A$ is an upper triangular matrix, then $A^{-1}$ is an upper triangular matrix, and the conjugate transpose $A^\*$ is a lower triangular matrix.

If $A$ is upper triangular and unitary, i.e. $A^{-1} = A^\*$, then $A$ is simultaneously upper triangular and lower triangular. The only matrices that are both upper and lower triangular are diagonal matrices.
Upper triangular matrices are closed under multiplication.

If you consider the elementary matrices corresponding to row reduction, all the upper right entries concern changing the upper right entries, so changing them back would use matrices of the same shape.

$A^\*$ is a notation that I have seen mean different things in different contexts. Does it mean complex conjugation here? If so, no elements get swapped so of course it would still be upper triangular.