I agree with IRstuff & others, if the goal is to keep cold beer cold for as long as possible, conserve the ice by draining the water to reduce the conduction/convection heat transfer from the internal wall to the ice. Another good reason is outlined in the fishy part below, changing "fish juice" to "residue from unwashed hands".
OTOH, if the goal is to rapidly cool warm beer, leave the water in (to keep the convection/conduction to the bottle high)(note also that no drinkable beer comes in a can) until the beer is cold enough to drink, then drain the water out.
Heat capacity: the heat (cold) capacity of water is 1/80 the heat capacity of ice, due to the heat of fusion (80 cal/g ice vs. 1 cal/g water). Until the ice melts, the temp. of the ice/water bath remains at 32 F/0 C, and the water contributes nothing from its heat capacity, since its temperature never changes.
Fish: keeping the fish out of water is best, but the more important reason (besides keeping ice icy longer) is that melting ice yields fresh, pure water. Fishies, even fresh-water fishies, have a saline fluid in their bodies. Soaking a cleaned (dead) fish in fresh water causes its internal, um, juices, to continually leach out of its tissues due to osmotic pressure (note that living fish have cellular mechanisms to resist this effect), the result is the fish's flesh gets soggy and limp, and its eyeballs sink, two signs that most people use to say whether a fish is "fresh" or not. Secondly, the leaching fish fluids make a lovely protein-rich bath for microbes to multiply in, resulting in your fish smelling fishy (the third way people determine fish freshness).
Say this 3 times fast: "leaching fish fluid makes fish smell fishy".
Grant proposal: I believe on the whole, the order of magnitude of drained vs. undrained ice retention time differences would be small, for an average-sized cooler on an average summer day. I further submit that everyone reading this post should pony up $1 US, and I will gladly conduct a side-by-side comparison test, with temperature-instrumented coolers and two cases of bottled microbrew; test protocol would involve removing one bottle from each cooler per 30 minute elapsed time interval, and recording temperatures of the fluids prior to their, ahem, disposal. The test would probably need to be repeated on a second day, swapping coolers, to avoid having a defective cooler skew the results, therefore make it $2 US.