It depends. I don't do so just for the sake of thanking or acknowledging - I figure I send them enough emails as it is, without saying 'thanks' every time.
But sometimes I do, yeah, for example if they offered immediate advice, or asked me if it was OK.

I generally email for 2 reasons: to share (issues that come up during the week) or report (doctor's visit) something we can then discuss in session. Or else for immediate (short) advice (such as medication) or questions. I don't like to do therapy by email, and I like pdoc and T to read and maybe respond "OK, we'll talk about it next time". Sometimes more in-depth has its place, but often that is enough. (I also email for rescheduling, which of course means we both verify)
Basically also 3 types of email:
*pure sharing/reporting: no real reply desired, I generally don't answer the reply
*emailing with, or ending the email with, a question: please answer my question. Depending on the answer or advice given (Or question asked back) I might reply
*hopeless sharing, = sharing when I'm extremely hopeless. I email to share/report, they will reply trying to help. It works. I will at least thank them, might also say I'll try to do as adviced, or comment on whether it has since gotten better of worse.

Honestly, my pdoc emails me after every lab (which is at least once a week). I'm not going to fill his inbox with a "Thanks" every time.