Heavy rifles are more forgiving of less than perfect shooting form and are easier to shoot accurately. Heavy barrels help make for heavy rifles. In my opinion this is the main advantage of heavy barrels. A quality lightweight barrel has the potential to be just as accurate with one exception. Shorter barrels tend to be more stiff. In order to get a longish 26-30" barrel as stiff as a standard 20-24" barrel it must be thicker.
If you are talking about long strings fired rapidly a heavy barrel will mantain accuracy a little longer before it overheats, but the inside of the barrel, where the rifiling is, will be just as hot. A heavy barrel also takes longer to cool down than a lighter barrel and could be a disadvantage under certain circumstances.
A very lightweight barrel can be amazingly accurate if the shooter does his part, at least for 3 shots, or if enough time is allowed between shots. I used to own a couple of heavy barreled target/tactical rifles, but sold them because the accuracy differnce between my lighter sporter weight rifles was just not that much greater. I never used them except at the range.
These are typical of a Remington Mt rifle with a pencil thin barrel. By taking about 1 minute between shots this is a pretty normal group. 5 shots, or rapid fire, would result in groups at least 2X that large.