Most of the systems call for a dedicated circuit. The typical draw on an electrical floor warming system is in the order of 10-13W/sqft, so might work, but generally, installation requires the manufacturer's instructions to be followed, and if it says a dedicated circuit, that's what you'll need. A high powered hair dryer might put it over the top..
You need to understand that when it comes to tile, it is not the size of the room that is critical, it's the strength of the structure. You need to find out that by measuring the unsupported length of the joists, their spacing, their height, and ideally, their species. There are load tables and a few calculators around that can help you there (check
www.johnbridge.com and use their 'Deflecto' calculator if you have a subfloor). Without knowing those specifications, there's no guarantee tile will survive. Many places in the SW are built on a slab, so deflection isn't an issue, although cracks can be (you won't know until you remove the vinyl)! Without under slab insulation, the response time and efficiency of infloor heating/warming won't be as good as you may like. They call the electrical systems floor WARMING, not heat. But, in a mild climate, it may provide the heat you want.
Others have reported, depending on the green rules that may apply, it may require a timer on the thermostat limited to 30-minutes at a time. This won't work well for heating. One installer in CA had that issue with the inspector (and the state codes). Have not looked at AZ's rules.