Cells need oxygen to function and clearly the cells in a zombie are at least partly functional. In the Walking Dead, do zombies need oxygen? Can they breathe? If they don't have functional respiratory or circulatory systems, do they absorb some oxygen through exposed tissue?
No. In pretty much any mythology (at least where zombies are dead, not living 'infected') zombies do not need to breathe at all. There is no explanation given for how they function without oxygen, nor is there an explanation for where the energy they use comes from.
Zombies have been seen to operate just fine underwater for extended periods, and zombie heads are still animate (unless the brain is destroyed) even without lungs. It is probable that a zombie could function in a vacuum, unless they were frozen stiff.
This is, frankly, a conceit of the genre. There's no real explanation, and I don't think there can be one. We do, however, know that they don't need oxygen to function.
Why do you assume their cells to be functional? In fact, in every single zombie representation I am aware of zombies don't heal. If their cells were working, I'd expect them to have normal healing powers but generally, when you cut a zombie, the cut stays open and you get those wonderfully gruesome images that so enhance our viewing pleasure.
So, while no explanation is given in any canon I know, the fact that they don't heal would point to their being animate but non-living tissue (whatever that may mean) and as such I would not expect them to need or use oxygen, no.
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.
And at least one buried (but still active) walker.
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.
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.