How to make Minecart Chests unmovable

Try summoning a Minecart Chest riding placeholder entity.

/summon MinecartChest ~ ~-1 ~ {CustomName:"Hidden Chest",Riding:{id:Chicken,Invulnerable:1,NoAI:1,Silent:1,Age:-2147483648,ActiveEffects:[{Id:14,Amplifier:0,Duration:2147483647,ShowParticles:0b}]}}

Note: This command is too long to paste into chat, use command block instead.

Added information about Invincibility - See Making mobs invincible (David Toh's solution for more details) and http://minecraft.gamepedia.com/Chunk_format#Entity_format.

You can do this by summoning an armor stand with a minecart chest as its passenger:

/summon armor_stand ~ ~ ~ {Invisible:1b,Marker:1b,Passengers:[{id:"minecraft:chest_minecart",CustomName:"{\"text\":\"Hidden Chest\"}"}]}

This is the command as updated for Minecraft version 1.13. Previously (prior to 1.9), this command was reversed; /summon chest_minecart ~ ~ ~ {Riding:{id:armor_stand}}. The Riding NBT tag was changed to Passengers in 1.9. A really cool resource to help you build these kinds of commands, which I used for this updated answer, is https://mcstacker.net/