We know that from at least Goblet of Fire and onwards, Perkins, the other guy that works in the Misuse of Muggle Artifacts Office, has lumbago (according to Wikipedia, a common disorder involving the muscles and bones of the back). But lumbago is not really anything to keep him out of work--we do have proof of this, as he meets Harry right before his hearing in OOtP. We also know that right before HBP, Mr. Weasley is promoted to the Office for the Detection and Confiscation of Counterfeit Defensive Spells and Protective Objects, leaving only Perkins in the Office. The scene does not mention Perkins. According to the HP Wiki, Perkins died in 1998. What happens to Perkins in between 1996 and 1998? What happens to the Misuse of Muggle Artifacts Office?
What happened to Perkins after 1996?
We don’t know. He’s never mentioned in canon after Harry meets him at the hearing.
However, note that HP Wiki doesn’t say that he died in 1975. The exact line is “(fl. 1975 - 1998)”. Here “fl” is an abbreviation for “flor”, a Latin word meaning “he flourished”. Typically it denotes the dates during which somebody is known to be alive, but their actual life may extend beyond those two dates.
It seems likely that Perkins was at least born before 1975 – it would be unusual for a 20-year-old to be described as elderly – but we don’t know anything about his death.
We don’t know whether he survived the Second Wizarding War. Although he worked in an unimportant Ministry office and was therefore “mostly harmless”, his association with Arthur and the Weasleys might have put him at more risk.
What happened to the Misuse of Muggle Artefacts Office?
Again, we don’t know. Never mentioned again after Arthur gets his promotion.
I think there’s a good chance that the office was shut down under Voldemort’s Ministry (or after Arthur got promoted – was anybody rushing to take over his job?). It was already held in low regard, and I can’t see Voldemort having much use for the office.
In a post-war time, it might have had more funding and recognition. When the Ministry began to fight back against anti-Muggle prejudice, it’s more likely that it was taken seriously. But again, I’m completely guessing.