Mike, I assume you are talking about water and not ambient air, right?
For water to mix a working solution of developer for film, I use distilled from a store nearby. They are selling a gallon of distilled for $.35 U.S. so I can actually afford it. Tap water is used for everything else, except the final rinse with Photoflo for film, distilled. Where I live, tap water is extremely hard (Tucson has been called the kidney stone capital of the world) but I use it without any problems for everything else. Your water source has a lot to do with filtration quality needed. Is it well water, city water or something else? Does it vary with the seasons as it does in some places? A tap mounted filter can help a lot for larger particulate matter and a magnet sitting on a plastic filter can't hurt for iron. A better system would be an under the sink unit if you really have dirty water.
For temperature control, I use a tempered water bath and tube development on film. Paper is a bit less strict on temperature, I just try to be consistent. Summer is the real problem. Ground water can be 90f or more at the tap in my darkroom, so I keep some water in the fridge for tempering. During the winter my darkroom stays at about 65 degrees in our "cold" weather, so it isn't a problem with heat until May when I have to run the cooler.
There are a lot of people who use tap water for all of their work. This is fine if you do everything the same each time and water quality is good. The only problem I can see is that if they travel or move, times for film may have to be redone around water quality.
I guess the best approach is to give it a try with tap water first and see what you get. If your film has things stuck to it when it shouldn't, a filter may be in order.