Why does the "almost no knead bread" recipe use beer? Can it be replaced?
The purpose of the beer (and the vinegar) in this case is to add some of the malty, fermenty flavors typical of longer-fermented or sourdough breads.  You can either leave it out and replace it with an equal quantity of water or use a non-alcoholic beer

The carbonation from the beer might add a little extra lift at the start to establish some air cells and work the gluten a bit, but with an 18 hour room-temperature bulk rise, that benefit would be negligible. In most beers (especially large US commercial bottlers) there isn't enough active yeast left in the bottle by the time it is drunk to do anything either.

The author of the original article (Cook's Illustrated #90, Jan 2008) added the beer and vinegar in order to add flavor to a bread recipe that already produced decent bread. 


  My bread now had tang [from the vinegar], but it lacked complexity.
  What I needed was a concentrated shot of yeasty flavor. As I racked my
  brain, I realized that beyond bread, there is another commonly
  available substance that relies on yeast for flavor: beer.