Moss takes its water in across its whole surface rather than through roots, so what is dissolved in that water matters far more than it would for an ordinary plant. The short version is simple: rainwater is kinder than the tap, and the harder your tap water, the truer that becomes.
Why rainwater suits it
Rainwater is soft, slightly acidic and free of additives, which is close to what most mosses meet in the wild. A moss on a woodland floor or a shaded wall has evolved to drink rain, taking up the little mineral it needs along with the water over its entire leaf surface. Hard tap water is a different drink altogether. It is alkaline and loaded with dissolved lime, and over a few weeks that lime raises the surface pH and leaves a chalky scale across the leaves, slowly turning a healthy green patch tired, pale and grey. Chlorine and chloramine, added at the treatment works to keep the supply safe to drink, do moss no favours either, and chloramine in particular does not simply gas off the way older chlorine treatment did.
When the tap is fine
If your water is naturally soft, straight from the tap is perfectly usable and you need not fuss. Much of the west and north of Britain sits on soft water; the chalk and limestone country of the south and east is where hardness bites. If you are unsure, a kettle furred with limescale is a fair sign your water is hard, and your moss will prefer rain. Where the tap is hard, save rainwater for the moss whenever you can. A single water butt off a shed or greenhouse roof gathers more than enough for a lawn, a wall or a shelf of terrariums. Letting tap water stand overnight lets a little chlorine escape but does nothing at all about the hardness, which is the bigger problem.
Storing and using rainwater
A covered butt keeps out leaves and mosquito larvae and gives you a steady supply through dry spells. Rain gathered off a mossy or dusty roof carries some debris, so let it settle and draw from partway up rather than stirring the bottom. Rainwater can go stale and grow algae if it sits warm and lit for months, so a shaded butt and reasonable turnover keep it sweet. In a hard-water district where you have run dry, boiled and cooled tap water is no help, since boiling concentrates the hardness rather than removing it; if you must use the tap, use it sparingly and flush the moss with rain when you can.
How to water
Little and often beats an occasional soaking. Misting keeps the surface damp without waterlogging, which is exactly what moss wants; standing wet with no air moving encourages algae and rot instead. Early morning or evening is the best time, so the moss is not left drying under midday sun. For establishing new moss, mist daily for the first few weeks, as set out in the growing guide; after that, in a properly shaded spot, the weather will do much of the job for you. There is more on adjusting the routine as the year turns in moss care through the seasons.
If the damage is done
A patch that has been watered with hard tap for a while and gone pale and crusty is not always lost. Move it into shade, switch to rainwater, and give it a few gentle rinses to wash the worst of the lime off the surface. Moss is patient, and a stressed cushion will often colour up again over a few weeks once the water is right, though a badly scaled patch may thin at the centre and take a season to knit back. The lesson holds for reviving any tired moss: get the water and the shade right first, then let time do the rest, as in reviving dried-out moss.