Which answer choice shows that the set of irrational numbers is not closed under addition

Since you didn't give the choices, I can only help with the problem. Then, you can select the correct answer.

A set is closed under a given operation, if all possible values in the set would produce the same type of number when the operation is completed.

For example: 
The square root of 5 plus the square root of 5 is two square roots of 5.
Both of numbers are irrational and they produce an irrational result.

Here is one example, like you should be looking for.
Irrational numbers are not closed under addition because:
Pi plus a negative pi would be zero. 
Pi and negative pi are irrational however zero is not irrational.
Therefore, the set is not closed.