My thought as to why the Overlook Hotel did not claim Dick Hallorann is because he was far too strong and a good a soul for the hotel to corrupt and manipulate, like it did Jack. Jack came to the hotel very very emotionally and mentally weak and vulnerable. The evil of the hotel that repeats itself time and time again, ever since it was constructed on top of Native American burial grounds (and was likely cursed since then), of course took full advantage of his state.
Some fans reason that Hallorann truly loved Danny and cared about him, and I'd have a tendency to strongly agree. Dick was the only character to truly care for Danny's wellbeing apart from his mother Wendy. Apart from Danny knowing Hallorann can shine like he does, there is also a strong level of trust, hence Danny telepathically calling Dick who is in Florida for help when he realizes what the hotel intends to do with Jack as its instrument.
But, getting back to Hallorann and his apparent immunity against the evil of the hotel, which he keenly senses is there, but is spiritually stronger than: Everything is energy in our world/universe, whether one believes in the supernatural or not. Jack came with a very bad energy to a place which was already supernaturally/spiritually condemned and was teeming with a highly malevolent energy.
Any medium or seer, even if they weren't energetically sensitive in the way Danny was specifically, would have felt/sensed this about the hotel and advised anyone sensitive to its energies to steer clear. But Hallorann, who I think basically amounts to Danny's guardian angel, even though he was murdered in the hotel by Jack, I believe was such a pure and good soul with a strong center, the hotel was simply not able to inherit his shine for its own evil purposes. Dick was untouchable. It is a kind of where-there-is-light-there-can-be-no-darkness kind of thing.
And also, Danny's shine was far stronger, even though because he was just a tiny little boy, he had not yet learned how to harness and control it. The hotel obviously wanted Danny's shine far more than Hallorann's. And it craftily knew a very young child, with little awareness of the world, would be fresh meat on the grill for the taking. Evil. Just absolutely hellishly evil. But this is King's genius.