First question, are you talking about the blue streak "slicks" racing tires?
if yes, then I have "driven" them on the street - - was ok for a lark, but there are extreme problems with this as an all-of-the-time thing -
first to your question about heat cycle, yes you can if you dirve them hard enough. If you plan to use these tires for street and then go to an autocross or track day, you might get away with it for one cycle thru a full track day and street use, but the next time at the track you will find they have "gone away". To get top performance, most race tires only get 4 (or for DOT race tires maybe 6) good heat cycles before they change dramatically. Also, slicks do not perform as you expect until you DO get some heat into them so you may just find that high performance street tires actually grip better in far more conditions.
Now for street use - the tires are so soft and roads are SO DIRTY that you pick up everything and then throw it up under your wheel wells with a tremendous amount of force!!! Even the rubber that is scrubbed off during "normal" driving can turn into problems. Fiberglass, even when coated underneath, does not respond well to this kind of treatment (stars in paint, web cracking etc).
Another problem is that they are trecherous in any wet - - even if you only drive on sunny days, you'd be surprised at how much water you will find (if only from people watering their lawns!), along streets, cross streets, when you make sharp rights and it's in the gutters, and when you accelerate = instant spin.
lastly, general roads are covered in dust (which is like water to slicks) and road grease from all of the cars (again like water). It's very nasty environment for true race slicks.
sorry for the ramble - hope some of this helps
dave