Are there any Terraria biomes that can't be transplanted to another location in the world?

Solution 1:

Pretty much all of the biomes (using Terraria's liberal definition of "biome") can be moved to another location, within certain limits.

The most extreme example is a Meteor crash site biome, which can exist anywhere, at any height. At the other end of the spectrum, we have the Dungeon, which cannot be moved at all.

Completely immovable

• The Dungeon, because it requires an unsafe background wall below the Dirt Wall depth (cannot be placed by the player).
• The Underworld, which is restricted to the very bottom of the map
• The Sky, which, like the Underworld, is just there at a certain altitude.

Mostly immovable

• The Ocean is not completely immovable, but very inflexible. It is both restricted to a certain height (above the surface level), and can only exist within 250 blocks from either edge of the map. You can create an artificial Ocean in the middle of the map, but that won't spawn Sharks, nor will it grow Corals. Creating artificial Oceans that sort of float above the real one does work; I have actually done so to farm Corals.

Solution 2:

In 1.0.6 they made it so placing dungeon walls and bricks were able to create 'safe' housing. I am not sure if this removes the ability to transplant the dungeon biome but it does seem to imply that. Note that a dungeon requires depth as well as the right blocks however.

Which brings me to my next point. The only other challenges to moving a biome are the ones that are elevation based as there is no way to get around that requirement. For example you can not get harpies to spawn at ground level just by digging a wide open space.

Other areas you mentioned above though? Yeah, no problem. You can create a new jungle just by making one, you can indeed get sharks to spawn by making a large enough body of water (Which is much easier to do with the new mechanical items added to the game).