To all those that aren't yet convinced the Heat have serious coaching issues (cough*HIF*cough): this is for you.
For me, the biggest problem is that the Heat's defense doubles, triples and quadruples players unnecessarily, leaving opposing 3p shooters wide open in the corners, in the process.
This is something for which Pops would kill you with his bare hands. The Spurs NEVER EVER EVER EVER leave open shooters in the corners. They'd rather give up a contested 2 than an open 3. And one might argue their system had some success over the years.
Exhibit 1:

Here TJ Ford beats House off the dribble, and the Heat's defense reacts by QUADRUPLE-teaming TJ Ford, leaving Granger and Dunleavy open in the corners. Dunleavy makes a 3 pointer.
Why would you need to QUADRUPLE team TJ Ford is beyond me. One guy should be enough to help House and either take a charge or block/alter Ford's shot.
Exhibit 2:

Again, unnecessary triple teaming of Rush opens Posey for a layup. Rush was not in a position to warrant triple-or even double-teaming.
You will notice that the Pacers are playing a small line-up, with Solomon Jones at C, while Spoelstra keeps Z on the court, instead of using the opportunity to play a quicker and more athletic line-up that could put some points on the board in a hurry: Wade, Lebron, Bosh, Jones, House.
Exhibit 3:

Again, basketball superstar TJ Ford triple teamed, leaving Rush open for a 3 (he didn't take it, but he was open). This time TJ didn't even get to beat House off the dribble before getting triple teamed.

You can see the play develop. Rush is now WIDE open, while the Heat is basically quadruple teaming, as Jones stays in the paint while his man, Dunleavy, sprints to the other corner. I guess Jones had to be there just in case TJ Ford bulldozed his way through Wade, Bosh and House.
Exhibit 4:

Collison gets in on the action as he was feeling neglected. Miami obliges and triple teams, although Arroyo was still in front of him. This leaves Granger open for a 3. If Wade closes in on him, Rush has an open path for a dunk or layup, or could set up for a corner 3. Or he could order lunch and some dessert and then shoot an open J.
Exhibit 5:

Collison gets triple teamed AGAIN. This time Solomon Jones is wide open a few feet from the basket and makes a J. You might call this a quadruple team, as Wade (4) also moves towards the paint, leaving Rush open. If the pass went to Rush he could have shot a 3 or Wade would have had to go too hard towards him, allowing Rush to beat him off the dribble and leaving Miami playing 4 against 5 on D.
Exhibit 6:

TJ Ford gets quadruple teamed. AGAIN. Granger and Posey are open for 3.
Exhibit 7:

How about a quintuple team for TJ Ford. Solomon Jones set a screen for him, so TJ Ford gets rid of Arroyo and is met by the entire Heat team. Notice how Arroyo instead of rotating to another man, keeps chasing TJ Ford, like a dog after his favorite toy. Jones is open for a J, but Posey is even more open for a corner 3. Which he makes.
Exhibit 8:

Granger is triple teamed on the baseline when a double team would have sufficed, leaving Dunleavy open in the corner. Bosh runs hard to cover him, but that means that when he gets the ball, Dunleavy can swing it to a wide open Posey. And if Arroyo moves in on Posey, that leaves TJ Ford open for a 3.
Just one more example of how exaggerating with the double and triple teaming puts your D in trouble and allows the opponent to keep you off balance just by moving the ball till they find an open shot they like - cause they can afford to choose which open shots to take.
Exhibit 9:

Granger is again triple teamed in a position where he's not that dangerous and while his man (Howard) is still in front of him. Here Posey is left open for a 3.
Again, we notice how the Heat play a big slow line-up and the Pacers are shredding their D with a smaller, quicker, faster line-up.
Another problem is how bigs are showing up when their man sets a screen. Normally you want your big to help and recover. But Z, Magloire and Howard are so slow that they can't really impede the process of the opposing PG, who gets around them easily. This results in the opposing PG having beaten Arroyo/House AND the opposing C/PF having a step on Z/Magloire/Howard. There are 2 men out of position and the Heat is playing 3 on 5. Before the defense scrambles to recover, the other team has already found and taken an open shot.
Exhibit 1:

Howard shows up on a screen by Posey. Notice the hand up in the air, when it should be down, trying to slow down TJ Ford. House gets picked and TJ Ford flies by Howard.

House and Howard are left in the dust. House tries to run after TJ Ford instead of rotating. Howard probably just tries to figure out what day of the week this is and what are those damn kids doing on his lawn.
Heat's defense overreacts with a triple team (while Lebron is also ready to step towards Ford). This leaves Solomon Jones open to roll hard to the basket and Dunleavy so alone in the corner that he's probably getting anxious and depressed.
Yet another issue is the play of Miami's PG. Especially Arroyo. Both him and House suck on defense, but at least House can hit the open 3 and help spread the floor. Arroyo on the other hand passes open 3s over and over again and, to make matters worse, is a ball stopper on offense. He likes to dominate the ball. When he gets the ball he feels the need to pump fake, take a few dribbles and then shoot or pass. This kills the rhythm of the offense.
Exhibit 1:

Arroyo gets a wide open look at a 3 pointer. Instead of shooting he pump fakes, allowing the defense to close in on him, takes a couple of dribbles and attempts a pass to Bosh inside, a pass that results in a turnover.
Exhibit 2:

Again, Arroyo is open for a 3. This time a corner 3. Again, he fakes, takes a couple of dribbles and causes a 3 second violation, as Z took up a rebounding position inside the paint, expecting a shot.
ALL THESE are proof of bad coaching, even if some are related to players' shortcomings. It's the coach's job to put the players in the best position to succeed, to emphasize their strengths and hide their weaknesses. On the Heat, however, it's quite the reverse.
* It's the coach's fault Arroyo plays with the 1st unit when he'd fit better off the bench.
* It's the coach's fault all these ridiculous double/triple/quadruple teaming takes place.
* It's the coach's fault slow bigs are left in the game when the opponent plays small ball - Miami should feast on small opponents, with Wade and Lebron as athletic and strong as they are. The ideal line-up would have to be (when everyone's healthy): Wade, Miller, Lebron, Bosh, Haslem or Wade, Jones, Miller, Lebron, Bosh and you can play it against small ball rather than a team with a lot of size.
I have nothing personally against Spoelstra, and I really wanted him to succeed. It would have been awesome had he turned into Phil Jackson 2.0. But he hasn't, and it's time for him to be replaced.