Just a few random observations here, no firm conclusions. I think alcor29 has a point about some inconsistencies regarding time, but I'm not sure if there's a bug.
First, on the fold.it website, the current UTC time is displayed at the top of most pages. That time always seems to be accurate. It's displayed without the UTC date, so a little mental translation may be needed.
(The time zone code used on fold.it is "GMT", which is OK, but maybe calling it "UTC" would be more correcter. It's been 50 years since Greenwich was officially part of the standard. GMT is still the official time zone in some parts of the world, but UTC is the reference time on which all time zones are based.)
Second, also on the website, puzzle expiration time is shown in UTC/GMT, but without the time zone code. Less than clear, and again, some mental translation is needed.
Other items on the website have the same issue. For example, forum and feedback posts show UTC date/time, but don't include the time zone code. It can be less than clear how recent a forum post is, even though the current UTC time is shown at the top of the page.
Third, in the the Foldit client, hovering on the puzzle title in the upper left shows the expiration time and the time remaining. The computer's time zone settings seem to be used, and the time zone code is displayed. For example, puzzles on my computer show expiration times in "CDT" or US Central Daylight Time.
Also, in the final 24 hours, the time remaining is constantly displayed in the score box.
For me, both the hover time and the final 24 time remaining seem accurate. If there's a bug, more information is needed, such as log.txt and maybe some screenshots. I'd suggest #bugs-and-feedback on Discord.
In general, I think using the user's local time makes the most sense. So the website should be consistent with what the client shows. For the web stuff, a little Javascript can convert the UTC time to the machine's time zone setting. Maybe this can be yet another requirement for the long-awaited new website.
For clarity, any display of a time should also include the date and time zone code.
A few spots on the website have a date only, for example, puzzles show only a create date. It probably makes sense to treat those dates as 00:00 hours UTC. I'm not really sure whether it makes sense to display a date and a time zone code without a time. I can't think of any examples where you'd see "03/18/2022 CDT", for example.
There's another issue regarding date and time format. On Windows, there are "Region" settings that control this aspect. I suspect that Javascript could also honor these settings, but not sure. I haven't done any testing to see whether the client displays of date/time use these settings.