Why does Hank throw Tuco's 'grill' into the river?

Yes they are all connected. Despite Hank's outward appearance of nonchalance, the incident with Tuco shook him a lot. It was probably Hank's first really violent fire fight and that left him suffering from a form of post traumatic stress disorder. He becomes more fearful and more fearful of further violence. Tuco's grill was a grisly reminder of all the fear he went through, which was something he didn't want to relive anymore. But fearful for his reputation as a tough cop, he outwardly accepts the souvenir only to dispose of it in secret later on.

Hank's secret reluctance to take up the position in the Mexican border is also another indication of this.

To add, I would also say that Tuco's grill represents the change that Hank refuses to go through. Hence, why he throws it, and for that matter, El Paso in the river. Heisenberg calls him and (if I'm not getting into spoiler territory), by extension death calls for him. So in a small way, I come to see Tuco's grill as both a grizzly reminder not only to the violence inflicted upon Hank, but the violence that he has inflicted on others.
Maybe the grill represents the tainted idea of being rewarded? I mean, Hank killed Tuco and yet, instead of having cuffs slapped on his wrists, people clap their hands on his back. "Good job, Hank", "Way to get that bastard!". But at the end of the day, Hank took a life, and I think that haunts him throughout the series. He constantly fears his own fate, verbally and emotionally (its why he has anxiety/panic attacks?) because he regrets his own thirst and actions for and towards violence.

The prior answers are good, but I wanted to point out a couple of other things. First, as one of the answers noted, the "grill" was not a trophy sought by Hank, but rather something that was given to him by his colleagues. Second, it seemed to me that Tuco was actually killed by the gunshot Tuco suffered during the strugggle with Jesse, fired by Jesse from Tuco's own gun before Hank arrived on the scene. Although Tuco was still alive during the gunfight with Hank, it was not clear that Hank even hit Tuco and Hank likely would have died had Tuco not been shot prior to Hank showing up. I think Hank felt that the trophy was not deserved.

I see the grill as a literary device which highlights the basic incompatibility between Hank and his wife. In a prior episode she steals the silver tiara, with no guilt, and gives it away as a gift. The tiara is a tiny silver semi circle, similar to Tuco's gold semi circular grill. In Hanks case however he is the recipient of this gift which eventually elicits a panic attack. Neither Hank nor his wife keep their precious metal arc - she blithely gives hers away, he casts his off with revulsion.

I believe Hank disposes of the grill because it is a constant reminder of his near death experience, as well as rage that Tuco wasn't the man distributing the Blue Meth.

Anyone who's seen Breaking Bad should know the answer, this is a fact not a guess, the reason why Hank throws the grill into the river was because it was a constant reminder of how close he came to death, and how scared he was in the beginning of this scene.

He approached Tuco as if he was Jesse Pinkman walking all cocky "Mr. Pinkman", he says with a smile on his face but then when he found out it who it was, s*** got real pretty quick, he turned that frown upside down because he knew how dangerous Tuco was and Tuco didn't even have a gun in his hands and he was already wearing a bullet in his stomach.

Hank, on the other hand, did have a gun on him but he was still shittin'. Hank's whole career he has the backup; his partner and his Squad of DEA soldiers that's why he always act so badass and tough because he basically has a military backing him up as he chases down skinny weak little meth heads, because in Albuquerque that's all there is, it's not hard up there. That gunfight he went through was a real traumatic day in his life, it was the first time shit ever got real and he couldn't handle it that's why when he got promoted to go to El Paso he knew that's where the real gangsters were, and he started having panic attacks, he couldn't handle it and that's why he got into a fight at the bar. He was trying to toughen himself up, he was embarrassed and ashamed that he really wasn't the tough guy he thought he was.

I can't tell you how many hints and clues there are in scenes where they straight out reveal this out right, when Skyler asks Marie how's Hank doing after the gunfight Marie replies you know Hank he's indestructible or something like that, this is a no-brainer I can go hours explaining evidence that supports this fact.

All answers on this thread are completely wrong. The correct answer is as follows:

Hank throws Tuco's grill into the river because Tuco was not the main man behind the Blue Meth. Tuco was small potatoes and killing Tuco does not warrant a Trophy so Hank tossed it out. Hank is a man driven by the success of getting his man no matter what the cost.

NO! Hank does not suffer from post traumatic disorder. The episode where Hank suffers a panic attack in the elevator - Hank knows he hasn't got his man yet, he just can't let it go, in other words he is VERY deeply disturbed/angry that he KNOWS the case isn't closed. Remember when Hank has a another panic attack while driving? He was absolutely FURIOUS that Walt was the guy he had been searching for all along. Don't forget what happened to Jesse when Hank found out he made a false call about his wife being in a car accident (poor Jesse suffered the wrath of Hank).

In other words Hank's "Panic Attacks" is Blind Rage bubbling up and exploding on him. Hank threw the nice Janitor (Hugo) in jail, he was prepared to throw Skyler behind bars and he was ready to bury Walt. The reason Hank doesn't want to go to El Paso is because he has unfinished business with the Blue Meth, he HAS TO GET HEISENBERG and only then will Hank move on. Hank is a man on a mission, don't fuck with Hank when it comes to getting his man. <---- THE END.

(This is the proper answer to why Hank threw Tucos' grill in the river) Oh the episode in the elevator, injuring himself while bottling beer, getting frightened by loud bangs that turn out to be beer bottles exploding. Is this all interconnected? No these things aren't really interconnected. The exploding beer bottles shows that Hank is ready to throw down with firearms.