Minor edits removing salutations, thanks-in-advances etc are in principle a good thing, as other answers have said. And as other answers also say, they come at a cost, including reviewer time (if edits are made by a low-reputation editor so go through the review queue) and of bumping questions to the top of the active list (taking attention away from more recent questions).
The tone of the other answers, and comments underneath them, suggest that overall this editing practice is discouraged, particularly on older questions. I would like to write a more encouraging answer, but with some suggestions to such editors (the kind who on Wiki projects is often called a "gnome") that would make the practice more useful and less harmful.
I don't see why age of question should in itself be a barrier to a "tidy-up" edit, though I see others disagree with me. But I would encourage any editor to consider the visibility of the question. Tidying up a question which is seen thousands of times in a year is two orders of magnitude more valuable than doing so to a question viewed only tens of times in a year (you can get a sense of this by looking at the "Viewed" stats at the top of the page). This substantially improves the cost-benefit ratio of the edit, and on high-traffic questions I'd suggest is worthwhile even on questions that are several years old. I'd also suggest that removing salutations is a lot more beneficial than removing thanks or other sign-offs, due to the importance of preview snippets (e.g. in search engines) where space is at a premium and salutations can really eat into it.
As a courtesy to reviewers, and to help counteract the cost of taking up space on the active question list, do try to find other things to improve as part of your edit. If you deliberately searched for a remove-worthy phrase (common salutation, thank you, misspelling etc) then it's likely the question was posted by someone not familiar with how Stack Exchange works or who may have made other mistakes with their English. Either way there's a good chance that you can improve the question further, even if it's only by adding more appropriate tags.
On the other hand, please resist the temptation of trying to make yourself useful or grow your reputation by making many such edits in quick succession. This renders the active question list useless as it fills up with minor edits. The "cost" of your edits will be much lower if you reduce the rate at which you perform them.
As an edit reviewer, the following behaviours would likely lead to me rejecting gnomish edits, even if I accept there was good intent behind them:
- low effort, failing to make changes to other obvious errors,
- rapid editing so multiple edits are jamming up the review queue and would dominate the active question list,
- a focus on questions with low traffic that don't seem to deserve another turn on the main page (e.g. the question is actually quite poor, even if not quite close-worthy, or very niche to the needs of the particular questioner rather than of general interest).