I had the same question!

My understanding — relayed through one degree of separation from Daniel Griffin, the dendrochronologist who wrote the piece — is that the core samples are very long and thin, and care is taken not to injure the tree and to allow it to heal rapidly.

Here's a write-up from Carleton with more info about how the tree core sampling process works: https://serc.carleton.edu/trex/students/labs/lab2_2.html

Not having read the much more authoritative response above (or any such), I'll foolishly offer the one fact about trees I recall from boy scouts or outdoor school:

The core of a tree is much less alive than the bark—so much less, in fact, that if you walk in a circle around a tree scraping off a thin strip of bark and make sure to stop where you started, the tree will die.

Having googled this just to be sure, I also learned that trees can only lose up to about a quarter of a circumference of bark (in the fashion describe above) before facing mortal peril.

Cork trees are an exception to this, and are harvested all the way round without harming the tree.