How to test for a named chest?

Blocks don't keep names once placed. So finding a placed block with a name is not possible. Only items keep the name, once you shift their state all properties are restored to default.

Names are not stored in blockdata!!

But, a chest has a block entity associated with it that holds additional data about the block. The block's block entity ID is chest. As stated on the Minecraft Wiki:

CustomName: Optional. The name of this container, which will display in its GUI where the default name ordinarily is.

Lock: Optional. When not blank, prevents the container from being opened unless the opener is holding an item whose name matches this string.

Items: List of items in the container.

LootTable: Optional. Loot table to be used to fill the chest when it is next opened, or the items are otherwise interacted with.

LootTableSeed: Optional. Seed for generating the loot table. 0 or omitted will use a random seed. Both loot table tags will be removed once the items have been generated.

The name of a chest is stored in the CustomName tag. So, to test for it, run this on a superfast clock:

/testforblock x y z chest -1 {CustomName:"The Name"}