My Sony Alpha 6000 experienced extensive burn in from photographing the sun. The burn in happened either during experimentation prior to the eclipse, or during the August 21 eclipse. The burn in areas are the same hexagonal shape as the closed down aperture of the lens. It was a manual aperture Leicaflex lens with an E mount adapter.
I don't think the burn in happened during the eclipse, as I used ISO standard viewing glasses, with one side taped over the lens, during partial eclipse. Totality was not intense at all.
My stupidity at testing how well I could image the sun without a filter is undoubtedly the source of the burns. I shot at 1/4000 shutter speed and the lens stopped down fully to f22. The result was an almost black sky with lens flare instead of an image of the sun, so I didn't suspect a problem.
After the eclipse I shot pictures into a fog bank on the Oregon coast. Later, when examining these pictures I saw what I first thought was dust, but what I later confirmed were burn in hexagonal spots exactly the same in every post eclipse picture. The burn in considerably darkens the image in the spot but does not eliminate image information.
Years ago I shot with the Leicaflex lens on a Leicaflex using film, trusting to the focal plane shutter and the mirror to protect the film from light except during the shot. This formed a habit I knew better than to carry over to digital, but did anyway. With a mirrorless sensor there is no focal plane shutter or mirror protecting a sensor that is continuously exposed and activated. When I examine the spots, they are multitudinous and distributed all over the frame. Some of them are solitary and some of them are worm tracks. I can only conclude that the damage happened as I was composing the direct sun pictures, and the distribution of spots is due to relative dwell time in any one position as I composed.
There is one possibility of which I am doubtful but have not ruled out. I used that camera just before the eclipse for a lengthy panoramic shoot where the sun from two hours before sunset to final sundown was long in the full frame fisheye scope of view, with the camera on. The aperture of the full frame fisheye was between f8 and f11. I have yet to examine that lens for aperture type and size to see if it matches the spots. I don't think the fisheye would have caused the problem with the sun low in the sky, and the camera was on a tripod. The pattern of spot distribution strongly suggests a typical handheld jitter, just the way I used it with the 90mm lens.
It is interesting that the burn in spots are not round but hexagonal, the same shape as the aperture. I have yet to fully explain that phenomenon in optical terms. The rays from the sun are parallel, they converge to a point in the aperture, and the image is formed on the sensor in reverse orientation at a distance from the aperture equal to the focal length of the lens. Also, because of extensive lens flare, there was no direct image of the sun. If the aperture shape is a shadow cast by reflected light, that reflection had to have been cast by an element of the lens. Certainly, there was enough lens flare to support this hypothesis, a work in progress.
For now, I would warn that with mirrorless cameras, don't point them at a light any brighter than what you would look at with the naked eye. I strongly suspect an arc welding light would leave a collection of burn in spots where ever the position dwelled for a time. It is not enough to turn the camera off or use a lens cap assiduously. A mirrorless camera sensor is at risk whenever you can see an image on the screen. However, if internal lens reflections are the culprit, the risk may be greater with some lenses, particularly long focal lengths with internal flare. Or the risk may be unique to the particular classic Leica 90mm f2.8 lens I used.
I will probably let this go without much further research. During golden hour there's no risk, and I shoot at golden hour.