It's not so much running tubes in radials, it's running tubes in tubeless tires. running tubes in a tire that is not designed to have tubes in them can lead to premature failure from the tube and the inner layer of rubber of the tire casing chafing, which leads to heat buildup and sidewall failure. What causes this problem is that the inner layer of rubber and the rubber that tubes are made of are essentially the same material. Think of it as the same as rubbing your hands together. In a tire, when this happens, the rubber begins to break down. That weakens the casing, as well as the tube. once one or the other goes, the tire will fail.
It should be noted that all major tire manufacturers will void the warantee on any tubeless tire that has a tube installed in it should it fail.
