I have thoughts on this, but I want to emphasize that I can only speak from my personal experience and have absolutely no expertise to back me up. Take everything I am about to write with something way bigger than a grain of salt... maybe, like, a giant salt lick.
I have tinnitus, fortunately not too overwhelming most of the time. It's usually relatively quiet and ignorable and hovers around 3.8 - 4 KHz in both ears (estimate using a tone generator to get as close as possible to the ringing I hear).
I'm fortunate that headphones usually don't aggravate my tinnitus, although not listening too loudly probably helps. That said, there's an exception: I rarely break out my Grado headphones anymore because I've found that, for me, they do seem to cause my tinnitus to temporarily worsen. Why? I don't know for sure, but I have a guess. One of the peaks that many of Grado's headphones are infamous for is right around the frequency of my tinnitus. I've wondered for a while if higher volumes at the same tone can aggravate the issue. That's speculation on my part, but it seems to fit.
Meanwhile, I can listen to my beyerdynamic DT 770 headphones with no apparent issues, and they're known for having quite a peak around 6 KHz or so. Maybe if my tinnitus was around that frequency I would have a problem? I don't know.
These days I mainly listen with headphones EQ'ed to the Harman curve or sometimes with an additional high shelf filter to lower the treble by about 4 dB, just to tame any brightness a little. Seems to work well for me.
YMMV, of course, and I'm making no claims that any of the above has any basis in medical fact. Just my own personal experiences.
Replying to myself here for context.
I'm due for a hearing aid refresh and, in the process of researching a bit, discovered that the manufacturer of my current hearing aids, Signia, promotes something called "notch therapy" for tinnitus. As I understand it, they amplify the things that need amplifying based on your particular audiogram, but they "notch out" a narrow band around your tinnitus tone (assuming your tinnitus is tonal, as mine is). They claim this can provide long term relief of tinnitus.
I haven't found a lot of research on the topic yet, and most of what I've found seems to be sponsored by Signia, so I'm still approaching the concept with a bit of skepticism, but I'm intrigued. If a notch in the frequency response can help relieve tinnitus, then my hypothesis that tinnitus can be aggravated by a peak at the same frequency seems reasonable.
I found a couple of links
here and
here. (The first one looks like an as-yet-unfinished clinical trial. That last one is sponsored by Signia, but at least they don't hide it.)
I mentioned my beyerdynamic DT 770, which has a notch around 4 KHz, which might make it more or less perfect for me even without EQ (and I've always liked it anyway).
I usually listen on my DCA Aeon RT Closed with EQ, but I was just playing with adding a notch filter in Equalizer APO, and I don't actually notice it much, if at all. I may play with it some more.
Standard disclaimer: This is not meant to be medical advice, talk to your audiologist, YMMV, contents may have settled during shipment, void in Kentucky if it's a Wednesday during a leap year, etc.