360 runs hotter with installed games?

The system runs quieter and cooler running games off a HDD. Not having the game disc in there spinning at 10,000+ rpm contributes to less heat being generated. The HDD will constantly spin at the same speed regardless of whether it's used or not, and reading off it isn't going to generate substantial heat. I always install my games before playing them even the first time.

One explanation is that in games that use a lot of content streaming assets can be accessed far faster from the hard drive than the DVD Rom. Since most of the textures in games are compressed along with other assets the CPU and GPU have higher constant work load decompressing them. Also once those are decompressed and loaded it puts more strain on the system to render the game. Just a guess since the 360 always runs hot and accurate measurement is practically impossible unless you pull the thing apart.

I would always recommend installing games if you can the benefits are very noticeable and improve many things.

The system does not run hotter if you install your games to the hard drive.  I don't know what give you that idea, but you don't need to worry about it.  You're more likely to suffer a DVD failure than than a general hardware fault with the latest models.

I've been running all of my games from my hard drive for the last two years without issues.

Installing your games means that the DVD drive won't be utilised as much, saving wear and tear on the drive motor and the laser.