In general water chemistry does not lead to dramatic 'off' flavors unless you really overdo things with the notable exception to that being the presence of chloramine which lends a plasticlike, bandaid or smokey flavor. One might, I suppose, consider the harsh, gagging, dry bitterness of the wrong kind of hops reacting with relatively small amounts of sulfate as an off flavor but some people load their beers with sulfate because they like this effect. Presumably they do not use noble hops.
It's also probably obvious that if you use lots of sodium chloride in your beer it will taste salty but note that this is appropriate for some beers.
More likely, though, improper water chemistry will result in beer that tastes dull i.e. all the flavors are muted rather than being vibrant, bright. This is the result of improper (high) mash pH. Conversely, I've heard people speak of the results of low mash pH as being beers where the flavors are too bright. I have personally never experienced this.
Beers made with very low mineral content have a soft, malty quality whereas those made with high mineral content have a sort of edgey brittleness to them. Note that both of these qualities are appropriate in some beers and inappropriate in others.