not always. nothing can be generalized about the sum of two irrational number. counter example. x=(sqrt(2) + 1), y=(1 - sqrt20) then x + y = 1, rational.
I'm assuming that you mean 'square root'. Yes, this sum is irrational. So are each of the two numbers alone.
A simple proof can be done by writing x=square root 2 + square root 3
and then...