It all depends on the bird, the rain and the environment!
Birds need to stay warm.. and they do it by trapping tiny pockets of air under their feathers. If those pockets fill with water - bird is going to get cold very quickly.
Smaller the bird, the bigger a problem.. since small birds lose heat more quickly and have limited energy reserves.
Hence, smaller birds might seek out shelter more readily than larger birds, so that they can avoid getting wet and cold. They take shelters in bushes, in reeds, in nesting cavities.
But taking a shelter does not work for long - birds also need to forage continuously to get energy. You will notice that, birds may suddenly disappear when rain first starts but if rain continues for long; birds starts to reappear. "They just cannot afford to stop foraging!".
Installing bird feeder at your garden or terrace under a shelter would give high chance of birds getting attracted to it. smile emoticon