Now, maybe he chose Slytherin, because he thought Malfoy was right about some wizards being better than others. Maybe it's because they hate muggles, and he hates muggles due to how he was raised. Now, dunno if Snape would still hate (or act like he hates) Harry, cause his in his own house, and friends with Malfoy, who is Snape's favourite. Maybe Harry becomes the favourite, or maybe it's still Malfoy. Anyway, I'd imagine that if he'd followed Malfoy's advice and gone for Slytherin, the two of them would get along. Possibly if he saves Ginny, he can swap Slytherin friends for Gryffindor friends at some point, but it's much more awkward because they don't share a common room and all.
IMHO, this would cause the most problems. The house is full of evil people that are against him, instead of a support base that he really needs. Shame, no sitting round the Slytherin common room talking about how much they hate muggles (him with some serious justification), playing tricks on Gryffindor mudbloods and blood-traitors.
Anyway, because he's the hero, he'd have to stop Voldemort from getting the Philosopher's Stone, maybe with Malfoy, and their trusty but dim-witted friends Crabbe and Goyle. Of course, they don't know it's Voldemort as such until fairly late on, and it's really awkward when they find out, given his friends are Voldemort supporter's kids.
Maybe instead of keeping the Stone from Voldemort, they want to steal it for themselves, or they don't find out? Yeah, Harry only knew because Bane told him, and Bane might not want to tell a Slytherin. More than that, Harry was only in the forest due to his thing with Malfoy, if they were friendlier, they'd not have known who was really behind it. And as a Slytherin, Harry might not want to tell his friends who was really behind it, of course...actually, I like that idea. Trying to stop Snape when they are Slytherins seems a bit odd, though.
Second book...Harry has to save some girl he doesn't know (except he doesn't like her family) from being killed by his friend's dad? Um...though again he doesn't know who's behind it till the end of the book, but then again, if he spent the holidays at the Malfoy's house, he'd probably have met Dobby. Also, Malfoy wanted to know who was behind it all, so he could help them. Maybe Harry is working with Malfoy on this, but gets worried when he hears about the school maybe closing, and him going back to the muggles. Assuming he did save the Gryffindor blood-traitor sister of a bloke he doesn't like and a member of a family his friends don't like...hmmm...well, he can say he was doing it to keep the school open. Probably Ginny won't be the love interest, though, or it will play out really differently. I'd say he could maybe still use the sword of Gryffindor, because that's the house he should have been in...he just keeps this to himself. If he saves Ginny, then presumbly Slytherin and Gryffindor start being able to get on a lot better, unless Slytherin really hold it against him, in which case he needs lots of Gryffindor friends.
Book 3, maybe Malfoy doesn't annoy Buckbeak into huring him. After all, if his friend had ridden him, instead of his rival, he might have been feeling more respectful. If Malfoy is injured, Harry won't be near Hagrid's when the executioner comes, just before Sirius attacks Ron/Wormtail. Unless maybe he wanted to see Buckbeak die, and got caught up with Ron and Hermione when Sirius came. Riding Buckbeak to save Sirius probably won't have happened, either because Buckbeak didn't hurt Malfoy and is off somewhere else, or he did and Harry isn't happy with that.
Book 4, Winky doesn't use Harry's wand, because he's not with the Weasleys, unless he saved Ginny and they are friends now. Harry is the Slytherin Tri-Wizard Champion. Suddenly he's got the house's support. Maybe he's been a bit too friendly with a blood-traitor girl, maybe he's spoil Lucius' plan by mistake, but he's now giving serious glory and prestige to house Slytherin, so that can be overlooked. His housemates and their friends busily help him cheat. Big problem when Voldemort summons the Death Eaters, and some of them are his friend's parents. OTOH, there was mention of Harry being thought of as the next Dark Lord. Now, if he got into Slytherin house, defended the Philospher's Stone against someone using strange magic (as long as he doesn't say who was behind Quarrell), defended the school against a murderous basilisk that someone (if it is still Lucius, maybe Harry/Dumbledore didn't figure out who, maybe Harry did but made a deal with Lucius to keep quiet) had set loose on a prominent magic family (even if the are blood traitors)...maybe some people give up on the long dead Voldemort and put their future in Harry. And when Voldemort comes back, maybe it's too late and they are committed, and they don't dare go back. Up until the end of book 4, none of the Death Eaters knew that he'd come back, and he was not happy with them as it was. Not saying this would logically follow, just that something like that, IMHO, has to for the series to continue along anything like the same lines.
Book 5 Dumbledore has to keep Harry safe, which probably means the Order of the Phoenix. If Harry saved Ginny, they should be able to mostly get along. If he didn't, Harry is probably that horrible Slytherin kid nobody likes...until he saves Mr Weasley (assuming he does, which does seems likely...as long as he recognises him, he's not had nearly so much contact if he's a Slytherin). Voldemort might not have a Sirius to lure Harry to the ministry, but if Harry was friendly with the Malfoys, they might find someone that way instead. Maybe even Draco.
A lot of the DA were Harry's friends from Gryffindor. Unless he's still got Slytherin support, they'd be hard to replace, unless Gryffindor really goes along with the idea (in which case he can have both). This means he's in trouble when he visits the ministry, which Voldemort definitely has to get him to do, due to the security there. Maybe the DA doesn't get caught by Umbridge, though, especially if he's got Slytherin support (Harry always knows to expect the Inquisition when it comes looking for him), and if Slytherin sneakiness meant they don't all write their names on a piece of paper with "Dumbledore's Army" on it and leave it behind...though that meant Dumbledore took the blame for them, might have helped in the end.
If some of the Death Eaters rallied behind Harry as a new potential Dark Lord or from fear of Voldemort, they'd presumably play nice(ish) with Dumbledore (keeping any future Dark Lord plans to themselves). Possibly swap the Order with ex-Death Eater a lot of the time, but there'd be all sorts of disagreements. Maybe Harry saves someone else instead of Mr Weasley (if someone else was on duty when the snake turned up).
Book 6...if Malfoy is with Harry, the Death Eaters need another way in...though Snape just needs to kill Dumbledore. If Dumbledore doesn't get disarmed/recognise Draco as going to kill him first, though, Voldemort gets the Elder Wand, and so Harry has problems/dies at the end of 7, unless he can disarm Snape somehow before Voldemort kills him.
Oh, if Snape was nicer to Harry for being in his house, then his potions marks might have been better, he'd have known he was going to do potions this year, and never seen the Prince's potion book.
Maybe if Harry was feeling more Slytheriny, he recognises Scrimgeous might not have been wrong to want to give people the impression that the ministry had things under control, so people are less scared and Voldemort has somewhat less success.
Book 7 is actually much the same, IMHO, assuming Harry gets this far. It doesn't matter which house Harry's companions come from, as long as he's got some not rubbish ones. The house structures keep them seperate at school, not in a tent in the middle of nowhere. Oh, given that Snape just chucked it in a pond, Harry should be able to get the sword...strange men walking past ponds distributing swords is no way of judging what house you are in, after all.
If Harry can get and maintain Slytherin support, then him being in that house is a very good thing. Instead of being the stock villain house at worst, or not taking part in fighting Voldemort at best, Harry could bring the Slytherin onside, and show Slytherin they don't have to be rubbish villains all the time. Now, he'll need to be on good terms with the other houses, but if he saves Ginny in his second year, that's going to help a lot.