DNS does not require its own currency.
No, but it does require a serialization mechanism whose developers aren't actively hostile to it being used for information storage.
In other words, you can't use the bitcoin blockchain for this.
So if it isn't its own currency (and therefore pays no block reward) and you can't substitute the BTC blockchain, why should miners waste computing power on it?
If you are going to decentralise you'll need a proof of work / proof of stake mechanism.
The monetisation of proof of work such as a namecoin like system is one way to achieve this.
Other proof of work systems could potentially be used too, I'd be interested to see what gets proposed.