Nope. The alignment apparatus attaches to the wheels, not the tires.But the tires could affect the wheel alignment just like a bent tie rod would. I could see if the old tires are worn unevenly they would sit differently on the alignment rack's turntables then new tires. Align with the old tires and then install the new ones and it then may wander or pull to one side or the steering wheel may not be centered again.
OP, just get an alignment. Doesn't matter if it's before or after new tires are installed. How about get it done the same time as the new tires are installed by the tire shop? Maybe they will give you a discount since you are buying new tires from them?