if you have chlorine in your tap water, it boils off. no need to filter
if you have chloramine in your tap water it doesn't boil off.
However every batch of beer I've ever made was made using alkaline, high bicarbonate, chloramine treated tap water....and my beers taste great.
It matters far far less if you brew extract beers. Grain brewers need to care a little more about their water quality since it affects extraction of sugars in the grain...which was already done to make your can/bag of extract.
in a nutshell: you either don't worry about your water, and consider it an aspect that defines your particular recipes...or you really get into the science of it, and re-create water profiles from around the world.
as far as tap water filters, depends on what type of filter. a Brita filter removes chlorine, some bad chemicals, and off flavors/aromas. An RO/DI filter will strip your water down to basically PURE h2o that doesn't really help your beer out...you'd wanna re-constitute it to contain the right amounts of certain minerals.
if you find the taste of your tap water pleasing without a filter, I would use it for brewing.
Primary: English Mild
On tap: Pale Ale, Lancelot's Wheat, English Brown Ale, Steam Beer, HoovNuts IPA
Bottled: MOAM, Braggot, Raspberry Melomel, Merlot, Apfelwein, Pyment, Sweet mead, Cabernet
Gal in 2009: 27, Gal in 2010: 34, Gal in 2011: 13, Gal in 2012: 10