I live in Phoenix, Az, where it gets super hot and revives very little rain. My garage was built by the previous homeowners out of 2x4s and plywood. The plywood is very weathered and getting weak in that if you kick it in some parts, you can kick a hole right through. Would it be safe/advisable to place T1-11 right over it? Or, does the old stuff have to come down? I'm concerned that moisture could form between the old plywood and new T1-11 and eventually have growth of mold or something between the two, even though it's so dry here. Any advice?

Do it right once, and tear out the old stuff before you replace it. That will allow you to INSPECT your structure and make a decision about it's long term integrity. Also, before you run down to the DIY big box store and catch the big sale, research your project by seeing if that's the best material to use in your area LONG term.

Are you covering the plywood with siding?

I agree with talaniman... not doing it right the first time only adds work and cost when you have to do it a second time.

1. eventually have growth of mold or something between the two

Even as dry as it is, mold and mildew spores are probably already there. I'd tear the old off. Make sure to keep the T111 at least 6" off the ground, I believe most codes use that figure.
2009 IRC Protection of wood and wood based products from decay shall be provided in the following locations by the use of naturally durable wood or wood that is preservative-treated in accordance with AWPA U1 for the species, product, preservative and end use:Wood siding, sheathing and wall framing on the exterior of a building having a clearance of less than 6 inches (152 mm) from the ground or less than 2 inches (51 mm) measured vertically from concrete steps, porch slabs, patio slabs, and similar horizontal surfaces exposed to the weather.

There's 2 grades of T1-111. The high grade is expensive but makes good siding with nothing behind it.