Basically, what Juances said. Firestorm Citadel is a popular Tier 3 mission because of its ridiculous Crown output, and it also drops a lot of heat; this is the area you go to in the mission "The King of Ashes".

If you haven't yet unlocked this mission, there are others you can farm. Royal Jelly Palace (The Sovereign Slime) and Ironclaw Munitions Factory (Built to Destroy) output around a thousand crowns per CE; Royal Jelly Palace's Boss Tokens can be traded for Brute Jelly armor that can then be sold, to increase crown earnings, but to my knowledge, the Schemer Scraps you can get from the Ironclaw Munitions Factory Boss Tokens can sell for a lot on the Auction House. Your choice; they'll both give about the same amount of heat.

If you're good enough and well geared enough, you can try a Danger Mission. These are tough, but they give a ton of crowns for the 20 energy you spend to get in, as well as a crapton of prestige; however, they are very difficult and may require you to spend more.

Finally, there's the Arcade- specifically, Arenas and Danger Rooms. An Arena is a level, shown by a Shield icon on the Gate Map, that has you fight waves of enemies with healing in between. Doing all three rooms in an Arena will net you a lot of crowns and heat. Danger Rooms are similar; they're areas in the Clockwork Tunnels, Blast Furnace, Cooling Chamber, Power Complex, and Wasteworks that require 3 energy to access. They have difficult enemies inside, but defeating them all will get you a lot of treasure chests.

One way to get a lot of heat is to revive players; when you revive a player, you get 30% of their heat. Knowing this, it's best to go somewhere you won't die in if you want to get a lot of heat, so if you're only Tier 2, don't go into a Tier 3 mission, because you WILL die in there.

second half/stratum of the tier gives more heat/crown (depth 4 - 7, depth 13 - 17, depth 24 - 28).
You can ignore the arena if it's not in those depths.