Here's a panorama using the same tool the article uses:
http://www.udeuschle.selfhost.pro/panoramas/panqueryfull.asp...
The furthest point from the top of Mt. Everest that you can see is 303km away, and it's tucked between two faraway mountains.
Edit: I'm sorry, but this comment is wrong. I used the tool incorrectly. In fact, longest LOS from Everest is 8th on the list, and is 455km long. I assumed that when the tool auto-completed Everest it would give me the peak, but it did not.
http://www.udeuschle.selfhost.pro/panoramas/panqueryfull.asp...
In fact, it looks like every item on the list is between two points that are each farther away than the other's horizon.
That raises the question of among all places on Earth where the longest line of sight is to the horizon, which has the longest line of sight?
Western peaks in the Andes are perhaps 150-200km from the sea. A tall peak in that portion could probably see the ocean. Chimborazo (6263m) is ~215km from the sea as the crow flies, although there is a large valley ~100m above sea level in the way.
I don't think Hawaii is the best bet.
Next place to look is probably at Puncak Jaya in New Guinea, which is 4884 m, giving 249.7 km to the horizon. But there is enough other land in the general vicinity of New Guinea that there could be over-the-horizon peaks visible, so it would take some work to check.
http://i.imgur.com/F8SqEj9.png
Reddit discussion: https://www.reddit.com/r/MapPorn/comments/6ofr5y/locations_y...
http://www.ringbell.co.uk/info/hdist.htm
However, the distances in the post reach ~334 miles. So it's not that simple.
You'll notice that several of the entries are between mountain peaks, which is what is really going on. If you can imagine a mountain peak that is visible beyond the horizon, the distance will be greater than the distance to the horizon.