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Page 110 of the DMG lists a DC 5 Constitution saving throw for "Extreme Heat" (temperatures of 100 degrees Fahrenheit or more). However, it doesn't make sense to me that it would remain the same as the temperature increases.
How do I fairly determine a DC for "Extreme Extreme Heat"?
As a DM, you have the authority to set DCs and other things and the game was designed expecting you to use this responsibility:
Many unexpected things can happen in a D&D campaign, and no set of rules could reasonably account for every contingency. […] The direction we chose for the current edition was to lay a foundation of rules that a DM could build on, and we embraced the DM’s role as the bridge between the things the rules address and the things they don’t.
Since the granularity of weather extremes is obviously not important enough to the default game for it to bother with the complexity in the core rules, if you feel like it should be more granular, you should make it so. If you've got a sense that DC 5 once an hour is insufficient, that is the same sense you need to use to adjust it.
That said, since you're talking about extreme heat, not just extremely hot weather (the world hot weather record is "only" 134 °F), you may want to consider your room more like an oven that does direct damage every round or every few rounds, possibly with a saving throw, since 200 °F is literally an oven. A cool oven, but still an oven.
Post scriptum: You — or other readers — may be tempted to get engineer-y about the precise temperature — and if you and all your group enjoy that kind of realistic detail simulation, go for it. But if you're not, I advise not trying to calculate too carefully the precise effects of particular high temperatures, since the reality of heat management is quite complicated — and besides, D&D campaigns are more often in a genre were fighting on the lip of a pit of bubbling lava is non-lethal, making comparisons to real-world temperatures somewhat beside the point. Go with something fun and plausible that suits the amount of on-screen time this room will have, unless you're all into the details of thermal physics.
I believe your best option would be to model the effects of the room off of one of the Heat Metal Spells from one of the older editions of Dungeons and Dragons.
The way the spells are engineered, the longer your victims remain in the room, the more pentalties, and eventually damage they suffer.
Perhaps an increasing DC Constitution saving throw against the damage until they leave the hot room.
You are expected to make a judgment call. You can easily gauge and declare a hotter zone has a higher DC, using the usual chart of target numbers, but also realizing that if characters are required to make several rolls over time, that even low DCs will rack up failures.
Now if you want to throw a little more consideration into it, you can start looking up stuff on exertional heat stroke. This report on fatal heat stroke in sports has one case of fatal heat stroke in the 70 degree range with high humidity though most it lists are in the 80s to the 100s range. So, at least as far as the D&D5E rules are concerned, they're being pretty generous in making it a DC5 roll.
An interesting thing to look into would be sauna related injuries or deaths, for example this article points out in a sauna competition, the winner stayed in for 16 minutes over 230 degrees. Mind you, in a sauna you're naked or near naked, and not doing anything - far from running and fighting and the usual adventurer stuff.
A few years back a few people died in a new age sweatlodge retreat where the temperatures were at 120 degrees, though the lack of ventilation, number of people, and the fact they had been in it for 2 hours straight also contributed significantly.
D&D is not particularly great at specific simulation, and the reality is that our current science points to massive variances based on age, fitness, humidity and many factors we're not exactly sure on to totally model heat exhaustion/death. That is, you'll find a lot that once the internal temperature goes above a certain amount, we have significant problems, but how fast we reach that point compared to the outside temperature is extremely variable.
So, even with science, it's going to be an arbitrary judgement.
