I'm adding an answer to give some further criteria in addition to the answers of iant and Aaron, and address comments made elsewhere.
I personally would not edit out a thanks unless a) I can or am already doing something else or b) I'm reviewing a post in queue - be it first posts, late answers, or whatever.
When reviewing those suggested edits, I first look at whether or not anything else could be changed/improved when reviewing an edit that just removes thanks. The next thing I look at is the age of the post being edited. Within a couple of weeks, I would probably approve it. While I have no fixed time frame, if it's more than a couple of months and there's nothing else that can really be improved, I won't hesitate to reject the edit.
In part this comes from when I asked a very similar question a while back (which you may want to give a read): At what point, if ever, is an edit 'trivial' enough to warrant rejection? In particular I want to quote part of a particular answer.
As a moderator I lean towards approving any constructive efforts to
improve our site, especially by newcomers, but I will not hesitate to
reject a trivial edit on a months-old post proposed by a seasoned
user, especially if that user seems to be embarked on systematically
editing lots of old posts.
Recently I noted a huge influx of suggested edits removing thanks all coming from a single newer user. Some of them on posts that were years old. If I could improve the post at all, I did so. If not, I rejected every one of those edits (which was most). It was quite clear they were systematically editing posts to chase either reputation or a badge. After I recognized the pattern I was even more inclined to reject them outright, even if there were other things to be improved, because they were clearly ignoring everything else wrong with the post and just deleting thanks/signatures. Note I did/now have the option of Reject and Edit, where it used to be just Improve. It appears that same user is back after likely being prevented from suggesting edits due to too many rejections.
Once you reach a certain rep level, your edits are no longer reviewed (and you do not receive rep for them). That might be a more appropriate time to make such edits, when you aren't tying up the review queue (though the whole point of the review is to teach you what is and isn't a good edit, something both editors and reviewers should keep in mind). It does still bump the posts though, so per the advice of others here and the linked question, I believe if that's the only edit that can be made on an old post, it isn't worth it. Otherwise I would be very tempted to go through the 2,713 posts that result from searching on "thank" and remove that temptation for new users.
I wish that the system would check if the username or the string "thank" appeared in a post when it was submitted and popped up a warning that stated thanks and signatures were discouraged (as it states somewhat obliquely with 'chit-chat' in the Tour) to have the posted correct the issue. The system does do this with some other things, but unfortunately not 'thanks'.
