By removing dead flowers -and the immature seed pods they contain -you encourage the plant to direct its energy into building up strength for the following year rather than into producing seeds, which is what it would do in the wild.
Take care, though, not to damage the young leafy shoots that originate close by the bases of the dead flowers. Old-time gardeners used to say that the dead flowers come away more easily if you bend them towards the north. Oddly enough, this does sometimes seem to be true, but it’s hard to see that there could be any scientific basis for it.
Leave the dead flowers and their seed pods on the plant only if you want seed for raising new plants. Bear in mind, though, that raising azaleas from seed is worthwhile only with true species. Most azalea varieties are hybrids and won’t breed true from seed. If you want seeds, leave a few pods only, more will weaken the parent plant.