I've not heard of a yeast-free pizza dough so this question interests me.
Breads either have yeast or chemical leaveners (baking soda, baking powder) to make the bread rise, rarely both. I had to add the baking powder in the search or I'd get just yeast recipes. With that in mind, a search for both 'pizza dough' and 'baking powder' brought up this recipe:http://www.cooks.com/rec/view/0,1941,153172-247207,00.html
In poking around further I saw someone mention using Bisquick for the crust and I found this Bisquick recipe for pizza. http://www.bettycrocker.com/recipes/extra-easy-pizza/689039a4-8c43-4a9e-9d48-27b1b0dda2c3
And last but not least, a thread from Amazon with the same question. http://askville.amazon.com/make-pizza-dough-yeast/AnswerViewer.do?requestId=4742747
As for your yeast dillema there's two things you can try to prevent killing it from heat. The cheap way is to use tap water that feels no warmer than body temperature or slightly cool to the touch. For more accuracy try a candy thermometer but be careful of going over about 100*F as anything higher may kill the yeast.
If the yeast is old it's already dead. Storing it in the fridge or freezer will give it a longer shelf life. The date on the package is a guideline but the best way to see if the yeast is alive is to proof it first before adding all of the ingredients.
To "proof" your yeast means to add the water and yeast (and sugar if the recipe calls for it) together and let it sit. If you don't see bubbles after about 5 minutes then it's a dud. No bubbles means it's time to start over with a fresh packet of yeast. Either the yeast is old or the water was too hot and it won't make your bread rise.