The rules don’t support this as an approach to bonus damage
There simply isn’t any rule for anything like this; improvised weapons always do the same damage and aside from magic weapons with magic flames on them, there are no rules for adding heat or fire damage to weapons.
But the game has tons and tons of things it doesn’t cover; it’s impossible to cover everything, after all. So the question becomes,
Did the rules miss some real effect that we should make a rule for? No.
Everything you want to do to make a weapon better at being a weapon, is going to make it worse at being a conductor of heat. Weapons want to focus their impact on a small area, and hit that small area with a great deal of momentum. Heat transfer wants a large surface area in prolonged contact with wherever the heat is to be transferred. These two goals are mutually exclusive, and I doubt you could ever impart a significant amount of heat by striking with a weapon that’s been heated over a fire.
A bullwhip (the kind of thing used by Indiana Jones) damages the skin via the extremely-high speed of the tip when you “crack” it—the crack is literally a sonic boom caused by the tip breaking the sound barrier. The amount of time that the tip spends in contact with the surface is extremely small, and must be for the bullwhip to actually cause harm, since its speed is so critical. Thus, the amount of heat that can be transferred in this timeframe—which is vastly less than the 1 second mentioned in the question—is also small, and not worth considering from a game standpoint.
On the other hand, a chain can’t be used like a bullwhip; the only reason a bullwhip works like that is because of the way it tapers (coming to a very thin point at the end), which a chain doesn’t do. Also, I’m pretty sure the links themselves are going to prevent the proper kind of flexibility and tension that a bullwhip needs, though I’m less certain of that.
Instead, a swung chain is basically a poorly-balanced flail, and swung properly, there isn’t a whole lot of difference between a chain and a solid iron bar. In both cases, you have to achieve fairly significant velocity at the end of the weapon to cause damage. The struck surface must then absorb some of the weapon’s kinetic energy, and in the process may be pushed away and/or break in some fashion. The typical approach to achieving the proper velocity, as well as to keep yourself moving and attacking (which is important both offensively and defensively) is typically to “swing through” your target. If you do this, again, the actual time spent in contact is going to be limited—less than a second. The amount of heat you can transfer in that kind of time might, just barely, be enough to do some damage, but I tend to doubt it.
And a chain is particularly poor here, because even if you tried to focus on the heat angle, you can’t just hold it against someone. With an iron rod that’s red-hot on one end—say, the classic fire poker—you can always just jab it into someone’s gut and try to hold it there, or swing it trying to force them to block it with their arm and then press in. It will cause nasty burns pretty quickly. But you can’t do those things with a chain; you have to keep it moving or it will just flop to the ground.
Also, all of that’s assuming we’ve struck bare flesh. Striking any kind of armor—or even just reasonably thick clothing—is going to see most of that heat get absorbed by the armor/clothing and not the skin, greatly limiting how much effect it can have.
Where a heated chain would actually be useful is wrapping it around the target somehow—as a garrote, maybe as a bolo or lasso, or, ya know, as chains. Thus, I could see having the chain deal damage when used to grapple a target. In this case, what I would probably rule is that you deal damage as appropriate for an improvised weapon, but in fire damage, on top of grappling the target.
