In Miui ROMs, there are four options for rebooting a phone from the power menu:
What does "Hot Boot" mean, technically speaking? Are there any circumstances where a hot boot could cause harm? Are there any reasons for choosing to hot boot at all?
If it works like other "hot boot" apps and functions, then it basically restarts the Android GUI processes/threads. There was an XDA news post a while back after someone made an app for it where the app author described it as such:
From what I know it's sort of equivalent to restarting your X server in a Linux environment (like hitting CTRL + ALT + BACKSPACE, for example). I would definitely doubt that you'd cause any damage by restarting your graphical environment, especially considering other OSes do this pretty easily, but I suppose there's an outside chance if MIUI implements it strangely.
Typically the reason people use a "hot boot" option is if the UI has somehow gotten sluggish and would benefit from a restart but they want to save time by not restarting the whole system (maybe a memory leak in the UI or something of that sort). It could also be useful if you need to restart the UI to apply some sort of change, like a theme perhaps.