For starters, I believe it is very important to create a separation between rooting for a team to not win an NBA championship, and hating a team. In this case, "hate" may merely be in reference to "hoping they don't win", but many will take that term literally. I have zero interest in the Miami Heat winning an NBA championship, but I do not hate them.
I think I have an issue with what the Miami Heat did to the idea of competition. Though, I have a tough time articulating in this regard. I think, in basketball, I've grown to expect championship teams and contenders to be things that happen over time, with a sense of cohesion, luck, strategy, blood, guts, key signings, sound trades, and hundreds of other mundane battles one must fight.
The Heat collection seemed to cut through a lot of that stuff, basically ending up in a land of, "You know what? I would like to become a champion, so let's get a bunch of prime superstars and perfect role players together and just do it." It felt a little
too easy to become a champion. They had no idea how to play basketball together in 2011 and
still came a miracle Mavericks comeback away from coasting to a ring.
I think most people have likely come in contact with stacked teams throughout their lives and have known how lame it feels to be on the other end, particularly when the stacked team was created by a bunch of guys on their own, as if to say, "Hi, we're the best players here and we call always being on the same team together. It's going to be pretty great when we win because winning is all that matters." I've seen that happen on street courts and it's just as lame. Great, yes, four college ballplayers showed up and insisted they always be on the same team. Yes, you defeated everyone because you're better. Well done. But what about the spirit of competition?
It's also kind of like, in high school, we had a powerhouse in our conference who'd dominate us year in and year out. It sucked. They were our rival. But it was what it was.
However, if that squad was only that good because they'd collected the best players from each other team in our conference, even if it were legal, that'd be lame sauce to the maximum.
TL;DR - I think many recognize that those fellers overkilled it. That wasn't just improving one's chances. That was the greatest player since Michael Jordan leaving a team that'd won 66 games at one point to team up with two prime Dream Teamers to make things as uneven as humanly possible in their favor. I think it's tough for non-Heat fans, and outsiders in general, to get behind that.
I think that was a part of it for me too. I knew how great LeBron James was. As someone who does not judge individual talent and ability off of team wins and team championships, it was not very difficult to recognize his legend before he even reached age 21.
One of the biggest reasons I was able to be a Pistons and LeBron fan at the same time (while those squads were rivals) was because I felt James did things the right way. I thought his leadership, ability to dominate while remaining a willing and skillful sharer was something I'd been longing for for a long time (in the era of volume shooting superstars). I thought he was good enough to raise the level of every player around him, and I think that was witnessed in him leading solid (though not spectacular) teams to 66 win seasons and NBA Finals appearances.
Basically, I felt he did basketball the right way. It seemed he had a squad and said, "Alright, this is my crew. Let's do this." And I felt he was going to be the one to bring down the evil empires of big markets, advantageous free agent signings & trades, and selfish superhero ball. Instead of being the type of guy who'd have the take of, "Aw man, I got Boobie Gibson out there?!", he was more, "Alright, how can I maximize Boobie Gibson's talent? I'm taking him along with me."
So for him to go from that, to the complete opposite end of the spectrum, as if to say, "Yeah, I know we were competing for a ring and any team I play for is pretty much a competitor by default, but screw it, I want a mega team" was a stark, stark contrast to everything I dug about that guy. It saddened me more than anything else. I kind of felt he may have given in to the folks who have the strange habit of judging an individual player strictly off of his team championship number. I thought he'd be above that.