Answer: The mulch from tires has two drawbacks that come to mind immediately, one that it can potentially burn/smolder/leach chemicals and the other that it does not rot down to feed the soil over time which is one of the main benefits to using a mulch. And, surprisingly, this type of inert mulch is not particularly effective in preventing termites -- neither is gravel or stone mulch!
There are some mixed opinions about the safety of tire mulch, as discussed here in an article from the University of Minnesota extension. You may need to cut and paste the complete url into your browser.
If you maintain a clear edge between the mulch and your foundation, and there is no wood from your house touching the ground, you have a better chance of resisting termites (assuming they have been treated and eliminated from the prior infestation.) There are many factors involved in attracting termites. Moisture is probably the number one attractant. From that perspective, any kind of mulch will hold moisture beneath it, including the rubber.
If it would make you feel better, besides the traditional wood based mulches, you can also use mulch materials other than wood, such as chopped leaves, pine needles, cocoa shells, or half finished compost for example. All of these organic materials will benefit the soil and your plants.
You might be interested in the following article about termites and protecting homes, from the Entomolgy Department at the University of Kentucky. It offers numerous tips to preventing termites.
I hope this helps.
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