This should be in the Rulings forum and please use card tags:
Pestilence Demon (Since there is no "Demon of Pestilence", I assume this is what you mean)
Once an ability is on the stack, it exists independently of its source. Even if the source is gone, the damage is still dealt. It will do so using "Last Known Information" about the source. A clearer example might be Mogg Fanatic. Obviously the ability works but it is gone before the ability resolves. The same can happen here with the Demon and the damage will still be dealt.
Pestilence Demon (Since there is no "Demon of Pestilence", I assume this is what you mean)
Once an ability is on the stack, it exists independently of its source. Even if the source is gone, the damage is still dealt. It will do so using "Last Known Information" about the source. A clearer example might be Mogg Fanatic. Obviously the ability works but it is gone before the ability resolves. The same can happen here with the Demon and the damage will still be dealt.