You can get away with any tire, but if you're doing high wattage intervals, or short powerful bursts, a dedicated indoor tire won't slip as much, so you won't experience big power drops, if that matters to you...
I have both but only use my Insideride rollers, and it doesn't make much difference (they both slip the moment I move my weight forward or not pedaling evenly out of the saddle), apart from being more silent.
Oh ! And a trainer tire doesn't heat up as much as a normal one, at least that's what I experienced. I get more flats with regular tires on my trainer than outside on the road.
HOpe that helps,
Louis :-)