Things I liked:
1. Companions. Yes, Bioware always does these well, but Inquisition's are really top tier imo. Thought-provoking, compelling, sometimes likeable and sometimes not, they brought the world to life for me in a big way. My personal favourites are Cass, Solas, Cole and Dorian, but frankly they were all marvellously memorable in their own way. Huge kudos and thanks to the writers, editors, artists, modellers, cinematics guys and all involved in bringing these fantastic creations into the game. You rock 
2. Visuals and performance. Very easy on the eye, and (I realise this isn't true for everyone) DAI runs beautifully on my machine with everything on Fade-touched and resolution downsampled to 2560x1440. 2 crashes in more than 300 hours of gameplay! Way better than average for a Bioware game. Again, I'm aware of and sympathetic to the fact that this hasn't been everyone's experience, but I can really only testify to mine.
3. Character creator. I love it and can happily spend hours in it. Thanks for giving us Black Emporium so we can tinker with the CC in-game.
4. Crafting, armour and weapon models. First RPG where I've actually enjoyed crafting. Armour and weapons look good, and I've spent an unreasonable amount of time tinting and upgrading things. Bear in mind this is coming from someone who would quite happily see crafting done away with in CRPGs because I usually find it so boring.
5. Some fantastic moments in the main quest - The Mage/Templar quest, the whole bit from the end of Haven through to the discovery of Skyhold, Wicked Eyes/Wicked Hearts denouement, Here Lies the Abyss, the end reveal. All great. Here Lies the Abyss especially. I loved the Fade, I loved the temporary companions, and the scene that
, it felt like I was experiencing something more than cosmetic consequences to decisions I'd made in Origins. And one of three quests in the game that had decision-making moments where I had to stop and think for a while, the others being
6. Romances. I've only played Blackwall and Solas, but neither follows what people often accuse BioWare romances of doing - a couple of conversations, then a sex scene, then nothing. The Solas romance is particularly good because it seems so relevant to the storyline and the world, because it offers genuine additional insight into Solas's character, and because the content is so well-written. Congrats and thanks to Patrick Weekes for this.
7. Archery. I just love playing an archery based rogue, it's wonderfully good fun. It's fast and mobile with lots of different skills.
8. Judgements. I thoroughly enjoyed these. it allowed for some nice roleplaying. I'd have liked to see a few more later-game consequences to some of the decisions, but it was still a ton of fun to sit on my throne and pronounce sentence on people!
9. Some of the side content. Side content appears on both my like and dislike list. Dislike for reasons I'll state (and others have already stated), but there was also good stuff. Companion quests, some of the regional quests such as Crestwood, and a few of the optional dungeons (Still Ruins for example) were decently fleshed out and had good story/lore behind them.
10. War Table. Also appears on both lists. Some fantastic opportunities for roleplaying, and a few (such as the Sutherland quest chain) that resulted in a 'proper' mission, and some changes to the world. I would have liked to have seen more of this. I don't think enough was done with the idea, but I would vastly prefer to see the war table improved, iterated upon and revised, rather than shelved.
11. The setting. Obviously not just DAI, rather DA in general. With the release of TW3 (and I'm sorry for mentioning it, the last thing I want is to turn this great thread into another TW debate), there has been some conversation in various forums about what makes a compelling fantasy setting. I've heard various arguments which claim that, in order to deserve a seat at the fantasy table, a setting should be deeply rooted in real-world Mediaeval politics and attitudes, otherwise it lacks believability and the ability to maintain interest. I vehemently do not believe this to be the case, otherwise it results in an entire sub-genre of games where only straight white blokes ever get to play the hero while the rest of us are either ignored, sidelined or treated appallingly in-game. I wouldn't argue that such titles shouldn't exist for those who enjoy those settings, but I am immeasurably glad that DAI isn't one of them - that it takes more contemporary attitudes as an inspiration instead, and is a fantasy space where I feel welcomed both as a female gamer, and as someone who enjoys playing female and gay characters, and having them be the saviour of the day. So, thanks for that 
Things I disliked:
1. Lack of ambient atmosphere, particularly in populated areas. Kids running around, domestic animals, a sense of people going about their day. Val Royeaux, Skyhold, Redcliffe, that small village in Emprise, Crestwood, all suffered a bit in this regard. Felt a bit sterile, there just didn't seem to be enough going on to bring places to life.
2. War Table. Just not developed enough. The Elven Inquisitor chain could have culminated in a fantastic quest to help your clan, but just kind of petered out with no recognition or consequence.
3. Side content. Too much simple filler that doesn't add much or anything to the world, and the better stuff is too easy to miss.
4. 8 ability limit. I liked this in theory, actually. I thought it might force me to change skills about as I played, but this isn't what happened. I ended up just loading up on passives as much as possible, and getting little or no use out of numerous actives.
5. Playing as a mage less fun than DA2. No spell combos (I loved those in DAO), and not enough spell variety outside of elemental spells. Some really good individual spells and upgrades (static cage!), but too much of the good stuff from regular trees in previous games was locked behind specialisations (like the bomb spells being necromancer only).
6. Underwhelming and underdeveloped main villain. Cory was the first villain in a Bioware game I haven't had an emotional reaction to. I've experienced the spectrum from hatred, anger, frustration, betrayal, to regret, respect, sympathy, even responsibility for Bioware's antagonists, but I didn't think Cory got enough screentime to evoke any feelings. His voice and design were fantastic though.
7. Main quest felt short, particularly the last mission had no real build-up.
8. Power mechanic. I didn't hate this, but in my case I ended up with ridiculous amounts of power that I have nothing to do with. In other instances, players complain that they are forced to do content they don't enjoy just in order to get the power they need to progress the main quest. This wasn't my experience, clearly, but it seems that it may be too easy to end up with either too much or too little.
9. Hair in character creator disappointing compared to the lovely hairstyles in the concept art (braids etc). Given Frostbite's lack of moddability, there is also little chance of many new hairstyles being created by modders (and the limited mods available haven't worked for me since several patches ago anyway).
10. Party banter. Magnificent when it happened, but didn't happen much.
11. Mounts. Great in theory, but given that the party dynamic is such an intrinsic part of Bioware's games, and given that I was already struggling to activate party banter, I ended up not using them at all. If there was a way (in future titles) to allow companions to have mounts too (I know there's a tricky pathfinding issue there) and still trigger banter, that would be great. Otherwise, I'm happier on foot.
12. Frostbite's lack of moddability. I'm not a modder, but the fact that DAO (and DA2 to a degree) could be modded made some difference to me as a user and added to the longevity of both games.
Well, that was stupidly long, sorry. I love DAI overall and look forward to more dlc 