### TL;DR:  No, zombies don't need oxygen to survive.  We don't know how zombie physiology works, but we know that they don't breathe.  

In fact, we've seen a number of walkers surviving long periods of time underwater.

![enter image description here][1]


![enter image description here][2]

![enter image description here][3]

And at least one buried (but still active) walker.


![enter image description here][4]

It has also been established that a severed zombie head remains active unless the brain is destroyed.  This is relevant, because obviously, severed heads don't have lungs attached to them.

![enter image description here][5]

The zombies in The Walking Dead don't need to do anything, really.  They don't need to breathe, as these gifs show, and they also don't need to eat, despite the fact that they do  so all the time.  Their respiratory and digestive systems are rotten and useless.  They can't breathe, and they can't digest anything.  Since they *can't* do these things, obviously they also don't *need* to do them.


  [1]: https://i.sstatic.net/2q8nJ.gif
  [2]: https://i.sstatic.net/2FSJQ.gif
  [3]: https://i.sstatic.net/DrIoJ.gif
  [4]: https://i.sstatic.net/kzQzU.gif
  [5]: https://i.sstatic.net/TYBFb.gif