Here's The Tempest's character system:
The sorcerer Prospero is the ex-duke of Milan, thrown out by his evil brother Antonio, who is thrown by a storm onto Prospero's island.
Miranda is Prospero's daughter who doesn't know her past and falls in love with Antonio's son Ferdinand.
Prospero is helped by his slave Ariel, a spirit whom he's rescued from imprisonment (from a magical tree); he's hindered by his rebellious slave Caliban.
Once he accomplishes his aim, Prospero promises to abandon magic.
Here's the character system of The Dark Knight Rises:
The superhero Bruce Wayne is the ex-billionaire of Gotham, thrown out by his evil brother (from the fraternity, the League of Shadows) Ra's al Ghul/Bane, who storms into Gotham and makes it an island.
Miranda Tate (aka Talia al Ghul) is Ra's al Ghul's daughter whose past is secret and who pretends to fall in love with Thomas Wayne's son/Ra's al Ghul's adopted son Bruce Wayne.
Talia al Ghul is helped by her slave Bane, who she's rescued from imprisonment (from a giant pit); she's hindered by her rebellious servant Selina Kyla.
Once he accomplishes his aim, Batman abandons superheroics, leaving all of his toys for Robin.
So The Dark Knight Rises is not a remake of The Tempest, as you can see: for instance, there's no Tempest-analogue for Robin--no one takes up Prospero's power after he casts it off; and there's a slippage between characters in DKR that isn't there in Tempest. I mean, Bane occupies two positions in DKR, one for his supposed role as head of the new League of Shadows and another for his actual role of servant of the real head of the new League of Shadows.
(You'll notice I also left off several other named characters: the rival businessman Daggett and his servant-who-abandons-him, Stryver; Alfred; Lucius Fox; etc. With some work, we could find their parallels in the Tempest--scurrilous Daggett sure seems like the scheming Sebastian, hoping to overthrow his brother Alonso, while the helpful Gonzalo might be Alfred or Lucius, etc. But we have enough with the main characters to see the parallels.)
But when you take a step back from the movie, certain character parallels and reused (or repurposed) tropes do show up, especially as inversions: Prospero is confined to an island before the beginning, while Bruce is tossed off the (newly made) island at the beginning of the end (Act III); the confining tree (above ground) becomes a confining pit (below ground); etc.
There are enough echoes that I wonder if Nolan looked to The Tempest once he had decided that the ending for Batman was Prospero's ending.