In Deathly Hallows,
when Dobby died,
Harry wanted to bury him without magic:
“I want to do it properly,” were the first words of which Harry was fully conscious of speaking. “Not by magic. Have you got a spade?”
Is giving him a Muggle burial the most honorable thing Harry can do?
Harry wanted to honor Dobby by doing the hard work himself. He didn't view waving a wand and having a hole magically appear as enough of a tribute for the sacrifice that Dobby gave.
It's an easy way vs. hard way, and the minor sacrifice of digging a hole by hand is the least of what Dobby deserved.
To expand on krillgar's answer somewhat:
Dobby was a house-elf. The job of a house-elf was to do menial tasks to make the life of a wizard easier. While Harry was perhaps not as strongly disgusted by this as Hermione with her S.P.E.W. campaign, he was familiar with how miserable this life could be, since he had grown up in a household where he was given the majority of the chores and housework. He therefore understood how important Dobby's freedom was to him (it was Harry that orchestrated Dobby's liberation, after all). Inherent in that was the idea of treating Dobby as an equal (Dobby himself commented on this when they first met).
It's likely, therefore, that Harry felt that using magic to remove the hard work from the task of burying Dobby would be disrespectful, given that Dobby had spent so much of his life doing hard work to make the lives of wizards easier.
Warning: long answer.
Now, I know this would sound a bit odd (well, a bit might be too soft expression), but one of the things, that came to my mind after reading the question is from Dumbledore's speech at the end of Goblet of Fire:
“Remember Cedric. Remember, if the time should come when you have to make a choice between what is right and what is easy, remember what happened to a boy who was good, and kind, and brave, because he strayed across the path of Lord Voldemort. Remember Cedric Diggory.”
I am very well aware of the fact, that Dumbledore is talking about kinda different situation. About the time when you must reveal, whether can fight for something, no matter the circumstances, and you stand to your best beliefs or if you cowardly stir yourself from your responsibilities and you sell yourself to the other side ( if you let me simplify it).
Now, I believe, that Dumbledore was talking about a very deep concept of principle upon you can build your life. A concept of making everyday or lifetime decisions, of taking the burden of responsibilities. You may then find, that what can matter is:
I think Harry is this kind of person.
Together, as said in the answer above, it was improper because of Harry's feelings towards Dobby, who while sometimes too much, cared for Harry's good. whether it was :
Harry realized this, I think. And well, there was not very much he could do for Dobby now, there was after all, nothing at all, that he could do for him.
He might remember the Christmas, when he did not think of Dobby, although Dobby thought of him:
“Can Dobby give Harry Potter his present?” he squeaked tentatively. “ ’Course you can,” said Harry. “Er . . . I’ve got something for you too.” It was a lie; he hadn’t bought anything for Dobby at all, but he quickly opened his trunk and pulled out a particularly knobbly rolled-up pair of socks.
He might feel pity, that he would not be able to repay to Dobby (not only this time, but also the other mentioned above).
The last thing, merely symbolic, was to honor Dobby by digging the grave with his own hands. To actually sweat and at least for a moment work as Dobby did his whole life.
Doing something very uneasy way is often considered character building process. Since it takes much more time, you can think about it. Parents/teachers use this punishment to teach you, that some things you should not take for granted, that is:
to expect something to be available all the time and forget that you are lucky to have it
Aside from the part about Molly Weasley I do remember four more occasions:
“You will be polishing the silver in the trophy room with Mr. Filch,” said Professor McGonagall. “And no magic, Weasley — elbow grease.”
“Can’t you help us with these sprouts? You could just use your wand and then we’ll be free too!”
“No, I don’t think we can do that,” said Fred seriously. “It’s very character-building stuff, learning to peel sprouts without magic, makes you appreciate how difficult it is for Muggles and Squibs —”
“He says you’re to come to his office at half past eight tonight to do your detention — er — no matter how many party invita- tions you’ve received. And he wanted you to know you’ll be sorting out rotten flobberworms from good ones, to use in Potions and — and he says there’s no need to bring protective gloves.”
“Mr. Filch has been looking for someone to clear out these old files,” said Snape softly. “They are the records of other Hogwarts wrongdoers and their punishments. Where the ink has grown faint, or the cards have suffered damage from mice, we would like you to copy out the crimes and punishments afresh and, makingsure that they are in alphabetical order, replace them in the boxes. You will not use magic.”
or the typical writing lines
In the real world imagine having to write many page essay by hand, no use of computer. Or searching for the information in the book, no internet.
Those things done by hand are somehow more real. It is not easy, it takes a much more time, if you do a mistake (/accident - spilled ink), you might need to e.g. rewrite the whole page just because of that one thing. During the process you are almost forced to concentrate only on your work, it gives you plenty of time to think about the circumstances, that led to it.
Now, it does not need to be only punishment. I think Harry did it as a sacrifice of his time(well, not just time); remember, how he was in hurry to get the information from Griphook and Ollivander after the burial. But he stopped everything about the Horcruxes and Hallows to honor his friend, a friend who was in the eyes of the majority of wizards seen as a filthy servant (for being a House Elf).
He dug with a kind of fury, relishing the manual work, glorying in the non-magic of it, for every drop of his sweat and every blister felt like a gift to the elf who had saved their lives.
Harry, being a (relatively) recent newcomer to the magic world, probably still sees magic like most muggles would; as a easy way to do something that you could do for yourself. Even wizards realized that this was the case. I don't have my books in front of me, but there was a scene where Mrs. Weasley was punishing Ron with peeling potatoes and expressly forbade using magic.
It wasn't that using magic would be improper. Harry just felt that it would be lazy to do so and that it wouldn't give due respect to Dobby, who had saved him on multiple occasions.
