On Tuesday I reviewed my Harry Potter and the Deathly Hallows forecast, pointing out story elements and parallels to Prisoner of Azkaban. I'd mentioned noticing various elements that paralleled Goblet of Fire, as well. These parallels are probably greater in number and more significant to the overall story than the PoA ones.
The first chapter of both books open with Voldemort plotting with his confederates and end with an Avada Kedavra and the death of an outsider.
We see a 'Potter Stinks' Badge in DH when Harry cleans out his trunk.
No Quidditch for Harry in either book.
Viktor Krum and Gabrielle Delacoer (Fleur's sister) appear only in these two books.
Gregorovitch, the wandmaker, is first mentioned in GoF, and plays a rather important role in DH.
Important impersonations with Polyjuice Potion in both books.
The Trio visit the woods where the Quidditch Wold Cup was held, when they are on the run.
Harry and Ron have a falling out in the Autumn of the year stemming from Ron's frustrations with Harry and the task before him. They don't speak for an extended period of time in both novels, then reconcile without much fanfare after Ron sees Harry in jeopardy (The First Task, and the frozen pool in the Silver Doe chapter).
We have a bona fide Death Eater as the Defense Against the Dark Arts Teacher. (Barty Crouch Jr. as Moody, and Amycus Carrow)
Harry gets crucial information in a Pensieve memory just prior to his confrontation with Voldemort.
We have some really interesting parallels to the Three Triwizard Tasks in Deathly Hallows.
Harry has to dive into water to get Gryffindor's Sword. He is rescued by Ron, mirroring Harry's rescue of Ron in the Second Task. The second task is again reflected in the Snitch that Harry receives from Dumbledore. He has to figure out how to open it and decipher it's secret, much like he has to figure out the secret of the Golden Egg in GoF.
The First Task is parallelled in Gringott's when Harry and company have to retrieve an item guarded by a Dragon.
The Third task is reflected in Harry's search for the Ravenclaw Horcrux including the Sphinx-like riddle to get into the Ravenclaw Tower and the search of the maze-like Room of Requirement.
There are also numerous parallels to the run up and final confrontation with Voldemort.
In GoF the final confrontation takes place in the Little Hangleton Graveyard among Voldemort's dead ancestors, while in DH, Harry uses the Resurrection Stone to commune with his dead loved ones and they stay with Harry, giving him strenth. This is also reminiscent of the Priori Incantatem incarnations of James, Lily, Cedric, etc. from GoF.
When Harry confronts Voldemort in GoF and their wands lock up, the effect grows and is described as '...a golden, dome shaped web, a cage of light.' In DH Harry meets Voldemort at the remnants of the great domed web of Aragog the acromantula. In both cases they are surrounded by a ring of Death Eaters.
In GoF Voldemort comes back from his half-life existence more powerful than ever with the ability to touch Harry at last. In DH Harry dies and is reborn more powerful than ever, with the Horcruxes destroyed he is prepared to take on Voldemort for the last time.
During their duels in both books Voldemort is ignorant of some rather important wand information which leads to Harry gaining the upper hand. Both duels come down to Expelliarmus vs. Avada Kedavra with Harry making his escape in the first instance and his ultimate victory in the second.