To me, the most important morals are these:
First, people need to focus on what is most important in their lives. We can see this in how Harry, Ron and Hermione stop attending Hogwarts and go off on their own to try to get rid of the remaining Horcruxes.
Second, people need to sacrifice in order to do what is best for their loved ones. We can see this most vividly in how Harry has to die in order to actually defeat Voldemort. He does not know that he will actually live, but he is willing to go and face Voldemort even so. By his willingness to lose everything, Harry actually gains -- he is able to defeat Voldemort and save his own life and everything that he holds dear.
There are quite a few morals that can be discussed in connection with the Harry Potter and the Deathly Hallows and the rest of the series, but I would like to focus on one which is easy to overlook, and this is trust.
Being able to trust is a moral strength that might pale in comparison to such virtues as loyalty and bravery; however, the theme of trust is woven into the plot of the Harry Potter series, and it comes to the forefront particularly in the Deathly Hallows. Throughout his long search for Horcruxes, Harry often doubts whether his blindly following Dumbledore's instructions is the right thing to do. "This is not love, the mess he left me in", he tells Hermione. Nevertheless, he perseveres and eventually understands why the way Dumbledore acted was for the best. On the other hand, Dumbledore, in concealing some information for Harry, trusted him to follow through with his mission regardless.
Trust also plays a crucial role in another aspect of the story line developed in the series, which also comes to its conclusion in the Deathly Hallows: the relationship between Dumbledore and Snape. The speculation whose side Snape is really on starts in the very beginning of the Sorcerer's Stone and continues throughout all the books. At the end of the Half-Blood Prince, the readers are convinced that Snape has deceived Dumbledore and has been working for Voldemort. However, as Harry finds out after Snape's death, Snape has been Dumbledore's most important supporter in actualizing his scheme to defeat Voldemort. Dumbledore repeats several times throughout the series: "I trust Severus Snape." In contrast, Voldemort does not trust anyone, even his most faithful servants.
Even thought there are instances when trusting gets Harry or other characters in trouble -- such as when Harry relies on Griphook to help him break into Gringotts, and the goblin betrays him -- the overall message of the Deathly Hallows is that one cannot win a war fighting alone, and life without trusting others is not worth living.
Power can lead to corruption. Immortality never works. You can hide from death but you can not escape death. Love overpowers all. Just like a fairytale ending.
Putting your life in front of your friends and that obtaining power in the wrong way will make you fall.