At many points in the story, when Dumbledore asks Harry something, he gives him a piercing look. It makes me wonder if that is Dumbledore using Legilemency against him. Is he?

He certainly seems to find some information when giving Harry these 'piercing looks', but he still doesn't know about Harry hearing the basilisk in the Chamber of Secrets, so I would say no; he's just extremely smart, so he makes some educated guesses as to what Harry is thinking.

I would also say that Dumbledore would feel it is rude to intrude on Harry's thoughts with legilimency, which is likely the reason he gives Harry plenty of opportunity to tell him his thoughts of his own accord. Harry's free will would factor greatly into Dumbledore's decisions.

I disagree. I absolutely think he was using legilimency. There’s a lot of evidence to support this, but the best example is when he asks Harry, notably in Chamber of Secrets, whether there is something he wishes to tell him. The reason Harry can tell when Snape is reading his mind is because Snape is using magic to force him to think of those images and thoughts. Dumbledore is much more subtle, as well as an exceptionally gifted wizard, so he simply asks Harry, who is very honest and open by nature (which is why he fails to learn occlumency with many lessons while Draco learns it almost immediately when taught by Narcissa) automatically thinks of the things he would tell Dumbledore if he weren’t afraid to, such as hearing the voice of the Basilisk. It also explains Dumbledore’s general tendency to know a lot more than he logically ought to. I originally assumed he was just making good guesses, and he jokes in one book that his guesses (not the word that’s used but I’m blanking right now) are usually correct. (Maybe the word is hunch. I digress) but I’ve also come to realize that when Dumbledore is amused with himself it’s on account of a private joke. Again I wish I had a quote for this but I don’t feel like digging for one.

I agree with this answer that Legilimency isn't imperceptible, that the 'victim' knows what information is being extracted and that the example with Snape and Harry proves this.

We do have one hint that Dumbledore used Legilimency on Harry, however. Since it's the only instance in canon that suggests this explicitly I thought I'd add an answer to the pile.

Boiling with anger at Snape, his desire to do something desperate and risky had increased tenfold in the last few minutes. This seemed to show on Harry's face, for Dumbledore moved away from the window, and looked more closely at Harry, a slight crease between his silver eyebrows.

"What has happened to you?"

"Nothing," lied Harry promptly.

"What has upset you?"

"I'm not upset."

"Harry, you were never a good Occlumens -"
_{(Half-Blood Prince, Chapter 25, The Seer Overheard).}

Dumbledore's behaviour here is consistent with the use of Legilimency. He maintains eye contact and focuses intently on Harry. Ultimately, he's able to work out that Harry's lying to him.

Arguably, Dumbledore wouldn't have needed to use Legilimency to work out that Harry was lying to him. Harry was not in control of his emotions and was pretty easy to read once Dumbledore started observing him closely. Moreover, Dumbledore doesn't find out exactly what Harry is angry about, which he probably would have done if he were using Legilimency.

On balance, I don't think that Dumbledore was using Legilimency here. Nevertheless, Dumbledore brings up Harry not being good at Occlumency, which could indicate that he had used it. I don't think he did but it's possible to make an argument that he did based on Dumbledore's wording.

No.

Harry would know if legilimency was being used on him, as when Snape used legilimency on Harry ^{_{(all emphasis added)}}:

“Liar,” said Snape. Harry’s throat went dry. He knew what Snape was going to do and he had never been able to prevent it. …

The bathroom seemed to shimmer before his eyes; he struggled to block out all thought, but try as he might, the Half-Blood Prince's copy of Advanced Potion-Making swam hazily to the forefront of his mind.

And then he was staring at Snape again, in the midst of this wrecked, soaked bathroom. He stared into Snape's black eyes, hoping against hope that Snape had not seen what he feared, but –

Here, what Snape was trying to see "swam [...] to the forefront of his [Harry's] mind", so Harry was an observer as well as Snape.

This did not happen when Dumbledore gave Harry a piercing look; I guess he was just looking sharply at Harry.

Remember, Dumbledore was extremely smart and had a lot of information that Harry didn't. Chances are that he could guess anything he was a bit uncertain about simply by observing Harry closely and his reactions to what Dumbledore was saying.

I doubt it.

If Dumbledore wouldn't use Legilimency on Slughorn to learn what he had told Riddle all those years ago, instead relying on Slughorn being honest, why would he read Harry's mind instead of just asking him? What ZenLogic said about Harry's free will is extremely important.

You have to remember that Dumbledore had taught literally hundreds of students, so Dumbledore's "piercing look" would have been something he had used for decades. I believe it is simply reading the non-verbal cues of a child that is not yet in control of their emotions.

YES.

Then how did Dumbledore and Snape know Harry was there under the invisibility cloak? Dumbledore saw him and knew he was there in Hagrid's house. How could he not be like Queenie? Throughout all movies he suggests he knows what people are thinking before they say it, esp. Obvious with Leta Lestrange.