No.  We don't know how zombie physiology works, but we know that they don't need oxygen.  

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]


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