House,
I need to know the brand & model of your joists, 16" is not enough info. Even then it can be tricky to simply say yes or no. There are many factors including their proper installation. We'll wait for your answer and go from there. Also tell us the type of tiles, size and method and products you are considering.
As for the subfloor, I too recommend adding an underlayment. Make sure it's "underlayment" grade and NOT "sheathing". Sheathing is a cheap grade and is used vertically, outdoors maybe? Underlayments should NOT the glued to the subfloor, are always installed across the joists just like the subfloor and offset both ways. Also the thickness of the subfloor has nothing to do with the deflection of the joists, only the space between the joists, which is where most of the problems start.
Jaz