Everyone has a different opinion on this. If you don't rest a pipe, then it doesn't have a chance to dry out and will often (not always) taste "sour"; but not everyone notices the flavor difference and some pipes seem to need less rest than others.
If you want to see the difference, take two of your best smokers, smoke one every day for a week (don't rest it or rest it only 1 day). Smoke the rested one (untouched for 5-6 days) at the end of the week and compare. See if you don't notice a difference in quality of smoke.
Personally I prefer to let my pipes rest 3-5 days (5 days is preferred), but sometimes with pipes I really like I will only give a day or two.
Like six says everyone has a different idea on this. It comes down to how big is your rotation. If you can afford to let it sit for a week all the better. If your rotation is small say two or three pipes, clean your pipes well and wipe them out after every pipe. Put them in a rack where they can dry and you won'tndo permenent damage resting for a day or two. I would advise picking up a few cobs to bolster your rotation to give your briars more time to dry. Also Meers don't need to rest the same as briar so that also might be a good investment.
I think the general rule is to let it rest a day at least. The longer the better, I suppose. I have seen people say that giving a pipe an hour or two to cool down and dry out a wee bit is enough, but I'm not sure if that's enough.
Remember, way back when everybody smoked pipes, people would only own one pipe, or maybe two - they'd smoke the thing all day, every day, with rests overnight.
As a rule a pipe I smoked throughout the day, 4-6 bowls, will rest at least a couple of days. It's not so much rest as just picking a different pipe for a different day. The briars are all well broken in with a decent cake and do not seem to suffer if smoked more often. A new briar will always get a drying out period until nicely caked.
My meers never rest and are always ready to serve. I have two meers that get smoked every night, one or two bowls each. I've got a handful of meers and every now and then one of these seems to catch my eye and get smoked. Generally, the 2 most used meers never rest more than . . . 15-20 hours? That would be an average. The meers can go day after day with no long rest periods.
+1 Six! Sometimes I might smoke the same pipe, with only one day of rest between smokes, for a couple of weeks. Sometimes I might rest that pipe for a couple of weeks between smokes. Most of the time I can notice a difference.