How old must a baby be to be lifted under its arms? Is there any required age? Is this kind of baby lifting safe for a newborn or toddler?
The key thing about lifting under the arms is that it's harder to support the head and neck, you'd have to use your hands as support, rather than the crook of your arm.
So it's less about age, and more about head control. If your baby is able to hold her head up to look around, then it's fine to pick her up under her arms. If the minute you try it, her head slumps to one side, not fine.
Frankly, you'd be able to tell immediately that her head is not coming up with the rest of the baby; if nothing else, she'll be getting upset by this shabby treatment.
While @deworde's answer is mostly true, there is another aspect:
It is fine to pick your baby up under the arms at any age as long as you support their head.
I don't know any of my friends who haven't held their baby up this way many times. It can be a handy way to bring them close - it just requires a little technique, like @Dariusz mentioned. When they are small their armpits fit easily in the crook between your thumbs and first fingers, which leaves the other fingers available to cradle the head.