The Shining is set in the Overlook Hotel, which is located in the Rocky Mountains of Colorado. Some important things to remember about the hotel are that it is isolated due to the winter conditions, it is absolutely massive, and it is (of course) haunted. The Overlook itself is based on the very real and very historic Stanley Hotel in Estes Park, Colorado, a place where King stayed and is said to be the inspiration for the book.
Similarly, "The Fall of the House of Usher" is set in a creepy, old, secluded, and massive place. The narrator goes to stay with "one of the boon companions of [his] boyhood" (Roderick Usher, who owns the place) and is very taken aback by both the place itself and the countenance of his friend who is nearly as creepy as the house itself—something to keep in mind. In the house, the narrator states:
I felt that I breathed an atmosphere of sorrow. An air of stern, deep, and irredeemable gloom hung over and pervaded all.
Despite this, the narrator notes that
[...] all this was apart from any extraordinary dilapidation.
So, the house isn't necessarily falling apart, but there is just something off and creepy about it. The Overlook Hotel is similar; there isn't anything immediately visibly wrong with it, but there is something that feels incorrect, which we later learn to attribute to supernatural forces.
Likewise, Roderick himself is described as having
A cadaverousness of complexion; an eye large, liquid, and luminous beyond comparison; lips somewhat thin and very pallid but of a surpassingly beautiful curve; a nose of a delicate Hebrew model, but with a breadth of nostril unusual in similar formations; a finely moulded chin, speaking, in its want of prominence, of a want of moral energy; hair of a more than web-like softness and tenuity [...]
All of these details help establish a very similar feeling between Usher and his house. He is creepy and off yet not outwardly decaying or dying. In this sense, Usher is a manifestation of the house itself. This idea is compounded at the end of the story, when Usher dies and his house breaks apart and dies with him, as if the two are supernaturally connected.
This brings us to one of the major similarities between the two settings. Both the Overlook Hotel and the House of Usher seem to have a supernatural control or connection over the people who inhabit them. Throughout The Shining, Jack Torrence slowly loses a grip on his own sanity—as if the Overlook itself is driving him crazy. Both of these settings are personified as evil forces in this way, a unique literary technique that is a major connection between the two stories. Interestingly, this point was seemingly lost when The Shining was translated from book into movie; a complaint often lobbed against the film is that Jack Torrence appears insane from the start, which undermines the power of the hotel that the book made central.
So, in summary, there are two major ways of looking at the similarities between the Overlook Hotel and the House of Usher. The first is the obvious one: they are both big, creepy, gothic buildings that evoke a sense of the supernatural and deep, penetrating isolation. The second similarity is how they affect the people who inhabit them by slowly eating away and destroying both their mental and physical health.
Another fun thing to note is that comparisons can be drawn between The Shining and other Poe stories, such as The Masque of Red Death. Poe is one of the forefathers of modern horror, and it makes sense that many lines can be drawn from him to King. These connections can wait for another day and another question, though. Good luck!