According to AskAManager, thank you, or follow-up notes are a good idea:
And they’ve really done everyone a disservice by using the term
“thank-you notes.” These aren’t really thank-you notes at all, but
rather follow-up notes. After all, most interviewers don’t really care
if you thank them for the interview; they’re not interviewing you to
be charitable but rather because they might want to enter into a
business arrangement with you that they’ll benefit from. So your note
shouldn’t be as much about thanking them as it should be about
building on the conversation from the interview. They’re “follow-up
notes,” not “thank-you notes.”
If done right, they can help (if you're already a good candidate). Here's another quote from her site:
Here’s the deal with thank-you notes:
* If you’re not the best candidate for the job, a thank-you note isn’t going to change that. No one is going to hire the lower-tier candidate
just because of a thank-you note.
* If you’re the undisputed top candidate, the lack of a thank-you note probably isn’t going to stop you from being hired.
* However, when the decision is close between you and another candidate, a thoughtful thank-you note can tilt the scales in your
direction — especially if the note isn’t just a perfunctory “thank you
for your time” but contains substance that builds on the conversation
you had during the interview.
* A thank-you note contributes to the overall picture of a candidate. It’s not generally make-or-break, but it’s a piece of the picture. It
serves two functions: (1) It signals that you pay attention to the
little things and care about presenting the best possible face to your
candidacy. (2) It signals interest, by showing that you went home,
digested everything you learned in the interview, and concluded that
you’re still enthusiastic about the position. That can matter. Now,
there absolutely are hiring managers who don’t care at all about
thank-you notes. But that shouldn’t dissuade you from sending them
because there are also plenty of hiring managers who will tell you
that a thank-you note has swayed their hiring decisions. And as the
candidate, you have no idea which type you’re dealing with … so of
course you should send thank-you’s. There’s just no reason not to do
this very small, very quick thing that could impact your chances. Not
that it definitely will, but it could. So keep on sending them.
It's hard to do a better job in what she says about them, but I'll point out that she is a hiring manager herself, not just someone who gives advice but doesn't actually hire. Some hiring managers (answering here) have never received a thank you, and don't consider them necessary. But since others, such as the one quoted above, do consider them useful (in some cases), it would be worthwhile following up after your interview.