In the case of the GS-R which requires premium, it can indeed be bad for the car. You're car is calibrated for premium, including various sensors, computer chips and the like. It's not a good idea, but it can be done.
Long term affects though, would probably not be worth it.
And while I don't have a lot of knowledge (yet) concerning the acura integra, I do have quite a bit when it comes to the different types of gasolines and what they can do for, and to, your car. I did a research article on it about 6 months ago. And my recommendation is : Use what the manual tells you to use.
another recommendation is : don't buy gas from the cheapie no-name brand convenience stores. While technically, all gas comes from just a few different companies. There's a lot that can happen to degrade it's quality from that point, to the point that it reaches your car's gas tank.
Such as: Different companies add different combinations of additives to their gas. these additives work better and worse, depending on your car.
Also, these smaller convenience stores often buy gasoline that is slightly under-rated in octane (called: watering down). Federal regulations allow for this to some extent.
As well: Each tank has a filter that the store is supposed to be replaced 2 times a year. It's not uncommon for these stores to let the filter go for several years before changing them, which means you'll be pumping in very dirty gas that is going to clog your own fuel filters and injectors.
As for brand recommendations. I choose, in order, Chevron #1, Texaco #2, Amoco #3, And Shell last.
Both Chevron and Texaco use detergents in their gas, it's advertised and well known and in American Made cars, Chevron gas does improve mileage. Shell has never come clean over what, if any detergents it uses in it's gas, and I get the worst mileage out of it, as opposed to the others.
Mechanics in this area also recommend chevron as well. In fact, the only problem with Chevron is that it can be hard to find a chevron station in the southeast region. We have 3 stations that I know of in New Orleans, and I know many places in florida and the areas in between don't have a lot of them either.
Phew. hehe.
Anyways, use premium, Chevron or Texaco, both offer good quality fuel with cleaning detergents. As for the mileage, you'll have to see for yourself. Like I said, it works well in American Made cars, but i've found, like my wifes Kia, some cars, almost nothing you do will improve gas mileage.