"It does not do to dwell on dreams and forget to live, remember that."
While Dumbledore's statement above makes good sense, it wouldn't do Harry any harm to try to analyze his dreams a little more than he does. Often there are important clues in his dreams to what's going on. As a matter of fact, Harry's extra-sensory powers are not limited to prescience. Besides the dreams, he often senses the presense of another person even when they aren't visible to him, hidden in the dark (PA3).
Harry's dream diary assignment for Trelawney in his fifth year would have been ideal for him to become more analytical on the subject of his dreams, of course. Unfortunately, given Trelawney's preferred teaching style of reading students' homework aloud in class, Harry knew better than to make the contents of his mind public property by writing an honest assessment of his dreams for her.
When he was younger, Harry had dreamed and dreamed of an unknown relation coming to take him away (PS2). Although at the time he had no idea, this dream would come true beyond his wildest imaginings on his eleventh birthday (PS4).
After his first trip on the Hogwarts Express, his introduction to school, and the Sorting, Harry had a strange dream. He was wearing Quirrell's turban, which spoke to him, telling him to transfer into Slytherin. The turban got heavier and heavier and tightened on his head. He saw Malfoy laughing at him, who turned into Snape, whose laugh became high and cold. He awoke after a flash of green light. The connections to Voldemort are obvious: the turban speaking to him, the high cold laugh, the green flash of light. It's interesting that Malfoy and Snape are connected to Voldemort in the dream, one transforming into the other (PS7). This foreshadows Harry's dream in Divination class several years later, when an eagle owl flies into a chair and Voldemort is revealed there; the Malfoy family has an eagle owl (GF29).
After spending three nights staring at the image of his family in the Mirror of Erised, Harry began having nightmares of his parents disappearing in a flash of green light while a high voice laughed (PS13). Voldemort's voice is described as high and cold.
For several weeks after facing Voldemort for the second time in his life, this time at Hogwarts, Harry would wake up in the night in a cold sweat, worrying that the Dark Lord was coming back (CS1).
When Uncle Vernon had locked Harry in his room and barred the windows, Harry dreamed that he was on show in a zoo with a sign saying UNDERAGE WIZARD on it attached to his cage. He lay on a bed of straw being stared at by people and also by Dobby, who refused to help him, responding instead that "Harry Potter is safe!" Then the Dursleys were staring at him and rattling the bars of his cage. When he woke up, Ron was outside his window, shaking the bars to get his attention (CS2). Given what Dumbledore explained nearly four years later about the Dursleys' role in his long-range plans to protect Harry, note the association in Harry's mind with safety and being trapped with the Dursleys.
After winning the Quidditch match against Ravenclaw in his third year, when Harry used a Patronus Charm to ward off what he thought were a couple of Dementors, Harry had a dream. He was walking through a forest carrying his Firebolt, following something that was silvery-white, like his Patronus. He could only catch glimpses of it and when he tried to catch up, he heard the sound of hooves. He woke up before he had a chance to dream more. The Patronus he was following was his own, which took the form of a stag, which was also the Animagus form his father had taken. Of course, he didn't know either of these facts at this point (PA13).
The night before the crucial Quidditch match with Slytherin in his third year, Harry dreamed first that he'd overslept and that his team had had to use Neville instead and had lost (PA15). Revisiting this after Harry's problems with his dreams in OP, and in particular the lost prophecy, it's interesting that Neville is the chosen substitute, and that the reason Neville had been needed was that Harry had overslept.
After the dream about Neville having to substitute for him, Harry then dreamed that the Slytherin team was riding dragons and that he had forgotten his broom (PA15). Less than a year later, of course, Harry had to pass a dragon in the air with his Firebolt (GF20).
Falling asleep in the library studying water-related spells for the second task, Harry dreamed that the mermaid from the painting in the prefects' bathroom was laughing and holding his Firebolt over his head while he bobbed like a cork in bubbly water next to her rock. She taunted him to come and get it, while he said he couldn't, fighting not to sink while snatching at it (GF26).
The mermaid dream is interesting because at that point Harry knows from the egg clue (although maybe not consciously) that the second task involves the merpeople taking his most valuable possession and he has to go fetch it back; since he's thinking in terms of material possessions rather than risking the lives of living people, naturally the Firebolt is the symbol of what he values most, an interesting point when considering some of Harry's other dreams in which the Firebolt appears.
In Trelawney's Divination class, Harry fell asleep and dreamed that he was flying on the back of a huge eagle owl. He flew toward an old, ivy-covered house on a hill and entered an upstairs window. He flew on the owl along a passageway to a room which was dark because its windows were boarded up. He left the owl's back. The owl went to a chair with its back to Harry, out of sight. Harry saw two things on the floor: a large snake and Wormtail. Harry watched as someone in the chair tortured Wormtail with the Cruciatus Curse after telling him that his blunder hadn't ruined everything after all. When Harry woke from this dream, he knew it was significant and he immediately left to tell Dumbledore (GF29).
During the summer after his fourth year, Harry had nightmares for weeks in which he revisited the graveyard in which Cedric had died, and 'even when he escaped nightmares about Cedric he had unsettling dreams about long dark corridors, all finishing in dead ends and locked doors, which he supposed had something to do with the trapped feeling he had when he was awake (OP1).' Harry hadn't even committed the offense for which he had the disciplinary hearing at that point, let alone had he visited the Ministry and seen the door in question. Those dreams, of course, could have been caused either by feedback from his connection with Voldemort, or by his own unusual talent.
On Harry's first night in Grimmauld Place, he fell asleep thinking he heard 'others making their way upstairs...In fact, many-legged creatures were cantering softly up and down outside the bedroom door, and Hagrid... was saying, 'Beauties, aren't they, Harry? We'll be studyin' weapons this term...' And Harry saw that the creatures had cannons for heads and were turning to face him' and ducked. Cantering is something either centaurs or thestrals might be expected to do, and both are later involved with the Battle of the Department of Mysteries - centaurs in both the confrontation with Umbridge and the Fountain of Magical Brethren, thestrals as transportation to the battle (OP6).
Through the days in Grimmauld Place prior to his disciplinary hearing, Harry slept badly, and continued having dreams about corridors and locked doors that made his scar prickle, although he'd never seen the door in waking life until the morning of his hearing (OP6).
On the night before catching the Hogwarts Express at the start of his fifth year (also, unknown to Harry at the time, the night Sturgis Podmore was arrested), Harry had a troubled night's sleep. 'His parents wove in and out of his dreams, never speaking; Mrs. Weasley sobbed over Kreacher's dead body watched by Ron and Hermione, who were wearing crowns, and yet again Harry found himself walking down a corridor ending in a locked door' (OP10). On the surface, the crowns are a symbol of the newly awarded prefects' badges, but it seems significant that Hogwarts will be spending most of the year listening to 'Weasley Is Our King' - especially since Draco hadn't even written it yet. (He wrote it specially when Ron became Keeper.) Interesting, in that light, that Hermione would be crowned alongside Ron. Kreacher seems like a symbol of Sirius' ultimate fate in this context.
Next we come to a dream that Harry himself thought was complete rubbish: the dream interrupted by his vision of Voldemort's giant snake. Harry dreamed he was back in the Room of Requirement with Cho, who was accusing him of luring her there under false pretenses, saying that he had promised her 150 Chocolate Frog cards if she showed up (same as the number of points a successful Seeker scores in a match - Harry, Cho, and Cedric, of course, were all Seekers for their respective teams).
Harry protested... Cho shouted, 'Cedric gave me loads of Chocolate Frog cards, look!' And she pulled out fistfuls of cards from inside her robes and threw them into the air, and then turned into Hermione, who said, 'You did promise her, you know, Harry...I think you'd better give her something else instead...How about your Firebolt?' And Harry protested that he couldn't give Cho his Firebolt because Umbridge had it, and anyway the whole thing was ridiculous, he'd only come to the DA room to put up some Christmas baubles shaped like Dobby's head...Given the previous (GF29) dream's Firebolt symbolism, interesting that Hermione would say that Harry should give Cho the Firebolt instead.
The famous witches and wizards cards resonate with the wizarding portraits; the image of them being thrown into the air is reminiscent of the ending of Alice in Wonderland, as Alice screamed that they were nothing but a pack of cards as they flew at her, waking to find her sister brushing leaves off her. Harry would have reason to associate Cedric with the portraits, given Vi running around the portraits on the night of their selection as Triwizard champions.
Given Dobby's decoration of the Room of Requirement for Christmas, the dream's inverted decorations are a neat touch, but consider the stuffed house-elf heads decorated for Christmas at Grimmauld Place where Harry actually did spend Christmas later, although he thought at the time he fell asleep that night he'd be spending Christmas at the Burrow.
Early in his sixth year, Harry dreamed of Ron chasing him with a Beater's bat the night after he began realizing that his own feelings for Ginny were less than platonic (HBP14). This was followed by dreams about Ginny herself (HBP15), but these don't seem to require a Seer to interpret them.
After Dumbledore's death, Harry's dreams "were thick with cups, lockets, and mysterious objects that he could not quite reach, though Dumbledore helpfully offered [him] a rope ladder that turned into snakes the moment he began to climb..." (HBP30)