Dexter feels a pathological need to kill and his code is what Harry gave him to make sense of this need. If he gave information up to the Police about his victims and put them in to the justice system, he would not only be robbing himself of the kill which he so craves and needs to feed his pathological hunger, he also would not be able accept that they had received adequate justice in his own mind for the crime which they had committed.

He is meticulous in his choice of victims and ensures that each of them meets Harry's code. If they meet this code, then in Dexter's mind the only justice for them is death.

I can't remember every episode, but I am sure that he expresses at least once during the 8 seasons that it would be easy for him to pass one of his victims over to Miami Metro without giving himself away. But that he just couldn't do that to himself, due to the resultant lack of feeling that he gets from the whole process of investigating them, through to disposing of the bodies from his boat. But more importantly, because in his eyes they just wouldn't get what was coming to them in the criminal justice system.

Dexter is a psychopath and a sociopath, utterly lacking in empathy or care for other humans, except maybe his family. He really, really doesn't care about whether the innocent live or die, for the most part.

The reason why his "code" targets criminals, serial killers and sick twists, for the most part, was by the design of his father. His father knew that if left to indiscriminate killing on the whims of a sick person, many innocents would die, Dexter would eventually get hunted down, and it is his father that feels the need to protect the innocent.

The code was his (father's) way to not have to intervene and arrest or even kill his own son, while giving his son an outlet for his urges that can actually be a benefit to humanity, by making the targets those who deserve to get snuffed out.

Dexter's personal priorities are to sate his urge to kill, and to not get caught. The code is merely a tool to help him achieve both priorities, but it is not some kind of moral code born from the beliefs and morals within Dexter. He really doesn't have morals.

Dexter, himself, is not all that concerned with saving innocent lives, as much as understanding that targeting innocents reduces his chance of long-term survival. Do the degree that harm to innocents raises his profile and brings scrutiny to his activities, it violates his code. Their lives are immaterial.

Surely he can't be satisfied that his 'need' to kill is at the expense of innocent people being killed.

That is actually wrong. He has a real need to kill people. And he does so following the code that his stepfather set out for him. The 'Harry Code' however does not speak of him having to save innocent peoples lives.

As Dexter is incapable of feeling (and in the books at least only mentions having some sort of feelings for his daughter), he would not feel remorse or feel bad about the innocent lives that are lost.

His primary concern should be to not raise suspicion about his own investigations. When he informs the Miami PD, he would actually be risking part of his cover. Which is a terrible idea if you're a serial killer.

Perhaps more importantly, Dexter wants to kill the people himself to satisfy his need. He kills the 'bad people' primarily for personal gain, not to help the 'greater good' or whatever. He kills because he has to and wants to. Especially if a person has done something particularly bad (threatening his foster family or real family). Dexter will, in those cases, prefer to kill the person than to just have him captured.

Another thing to keep in mind is that the justice system would allow for people to go to jail and be released at a later time. It has happened (at least in the books, can't recall in the series) that Dexter kills someone who has been released. Thus killing them just seems to be the logical option.

TL;DR

I don't really find any reason why this would conflict with the 'Harry Code' of 'Not killing innocent people'. He does not do the killing. Aiding in the murder of innocent people is not the same as killing them. Keep in mind, Dexter has no feelings.

The other answers seem close to me, but not quite on target. Dexter follows the code out of a desire to not be caught/out of habit (by the time in Dexter's life where the show takes place). It's not hard requirement in the sense that there are no necessary, mechanical consequences to breaking it (nothing prevents Dexter from killing a random person beyond his choosing not to do so, for example).

Dexter (as presented in the books and show) will kill. It's a product of his psychology that he doesn't feel the reservations about killing, or the remorse afterwards, that "normal" people would feel/do feel, and also that he experiences an irresistible urge to commit murder.

When Harry recognized that Dexter would almost certainly become a serial killer he devised the code, the purpose of which is explicitly to make sure that Dexter does not get caught. One element of that (which is also an effort by Harry to salve his conscience about the entire situation) is that Dexter should make sure that his targets "deserve it", interpreted in the franchise as murderers that would otherwise go unpunished.

So it's not about saving lives at all, but about constraining Dexter's inevitable behavior in such a way that he won't be caught. Dexter doesn't fight for justice, at least not directly, he chooses targets, in the same way that you know you will eat dinner but need to choose one specific meal over another. Descriptions of broader morality as related to the code may apply to Harry, but they don't apply to Dexter who isn't working out a personal moral philosophy so much as managing behavior that he will, eventually, engage in, no matter what.

As to Dexter's behavior in later seasons (5+), I feel that the show became sloppier overall, and this could account for violations of the code. But it's also the case that a major plot arc of those seasons (particularly season 6, if I recall correctly) is Dexter modifying or abandoning the code for various reasons. If the code can be changed or altogether discarded at Dexter's discretion, it obviously cannot be a hard constraint on his behavior.

Alternatively, if you don't accept any of the above, Dexter is not perfect. He definitely directly kills people he later discovers to be innocent of any crime as well as people who, while unpunished criminals, are not murderers.