Does the hymen go away after puberty?

No. Some hymens are little more than a ridge so a post pubeertal woman might think that it went away spontaneously. However a hymen usually remains until it gets torn or ruptured.
The short answer:NO. The hymen is a structure that acts as a partial gate to sensative vaginal skin before the hormones of puberty thicken the vaginal walls. Before puberty the skin may breakdown with trauma & easily bleed & get infected. After hormonal changes this is less likely. After puberty the hymen may have a small or large opening. The opening will be stretched or broken during solo or partner sexual activity.
No. The hymen never goes away, it just changes. All women are born with one, and all will die with one. The idea that it is a flap that can pop and disappear is patently false. It is a fleshy round piece of tissue thatched all the way around the inside of the vagina with a hole in the middle. It's flexible and stretchy and does not pop, just changes.