I used a EMF detector app on my phone and measured the EMF output on my wired headphones. It measured around 10,000 milligauss on one part of both ear cups. Since wearing these headphones on my head for long periods of time, will this cause injury to my head or immune system?
The ability to measure something, with your phone app for instance, makes it far easier to worry about it.
Consider the mass experiment that is being done now, has been ongoing for the past few decades, of people all over the world (billions?) wearing headphones. They will have similar exposure to low value DC magnetic fields. There is another population, running into millions, of people who've had exposure to fields 1000s times more powerful, inside MRI scanners.
With all that exposure, you might expect that any harmful effects would have shown up by now, and they haven't. Which sits fairly nicely with there being no known way that magnetic fields can harm you. Interestingly, there are known ways that ionising radiation like X-rays can harm you, and precautions are taken to minimise exposure to both to patients and staff in medical facilities. Contrast the two approaches taken to magnetism and X-rays.
Unfortunately, me knowing that magnetic fields are harmless doesn't make it any easier for you to feel comfortable wearing headphones, now that you can put a number on the leakage field from the ear cups. It depends whether you're into conspiracy theories whether you'll ever accept reassurances from any source.
Full disclosure, I once worked on MRI machines, and was often bunged into the machine when we needed a test image. When the ladies in the lab were running the test, a man was sent it, and vice versa, and the image was always of the genital region. Oh how we laughed! For all that strong magnetic fields are difficult to work with (wipe credit cards, zap pacemakers, get fork-lift trucks stuck to the magnet (yup, done that), bend CRT traces off the screen, send steel wrenches flying through the air), damaging human tissue through direct action is not one of the problems.