Why was Bruce Wayne so harsh/rude when he asked for everyone to leave his party?

He decided that being rude was the fastest way to get people to leave.

If he had politely asked it is very likely that a lot of people would have hung around to try to talk to him.

Equally if he had claimed that he was busy it would still have been weird to kick everybody out. Bear in mind that he lives in a mansion and part of the point of living in a mansion is that you can have a big space for public parties and still have plenty of private space for yourself.

If he claimed that some businesses thing had suddenly come up the normal thing would be to say 'sorry I've been called away, enjoy the party'. It's not like he has to serve the food himself or turf people out of his bedroom so he can use his phone in private.

By being rude he removes the incentive for people to want to stay.

In terms of the meta-narrative this also represents the point where he stops leading a double life and the Batman persona becomes dominant over the Bruce Wayne one ie keeping up the facade of Bruce Wayne is less important to him than his activities as Batman so Bruce Wayne as a reclusive and unapproachable placeholder is just more convenient and means that he doesn't have to put much effort into 'pretending' to be BW anymore.

To put it another way this is the point where it is BW rather than batman which is the facade which is also the reason why his relationship with Rachel breaks down.

He needed everyone to leave for their own safety, but couldn’t tell them that.

He perceived (correctly) Ra’s Al Ghul as a threat to the safety of everyone at the party. He knew they all needed to leave immediately, or become collateral damage. He also knew that he couldn’t just say that because no one would believe Bruce Wayne if he said that, and he couldn’t reveal that he was Batman.

So he needed a way for Bruce Wayne to get everyone to leave. Bruce Wayne was a drunken, ill-mannered, overly-rich playboy that no one took seriously. No matter how seriously or firmly he tried to convince people to leave, they weren’t going to, or at least wouldn’t quickly. He had legal authority to do so, since Wayne Manor was his private property, but it would take a long time to convince anyone that he was seriously asserting those rights.

Therefore, he needed a “Bruce Wayne” way of getting everyone out. Well, Bruce Wayne was well-known as a drunken, ill-mannered, overly-rich playboy. Getting drunk and insulting everyone was easily believed, far more easily believed than anything serious he tried to do. The guests had very little difficulty believing this behavior was for real. And these were rich, proud people—not the type to stand around and be insulted. That got them moving more quickly than anything else he could have done.

Now, he probably does genuinely dislike these people. And he is focusing more on being Batman than on being Bruce Wayne. But ultimately those aren’t really factors here—his sole goal was to get everyone out of and away from the building as fast as possible, and deemed this the most expedient route to doing so. He might not have cared very much what these people thought of Bruce Wayne, but even if he had, safety was the goal here. He was saving those people’s lives. That’s all that mattered.

When Bruce recognizes Ra's Al Ghul is back, he realizes he needs to end the party and get rid of everyone.

Bruce Wayne actually wanted everyone to leave before he even joined the party, just after asking Alfred if Fox was still there. With Bruce having just brought in Rachel [who had been poisoned], and with Fox being there, Bruce was more concerned with being Batman than he was maintaining the Wayne name and image. This is the argument that he and Alfred have just before Bruce joins the party.

What was the point of this? Was it because he was drunk?

No, he wasn't drunk, though he did play the part of one so that he could be perceived as a drunken jerk, thus allowing him to speak so rudely. Bruce wanted everyone to leave immediately, and so, throwing out insults and directly telling them to leave (and that it's not a joke), was his strategy.

Or was it because he genuinely does not like these people and do not want them to return?

This is much more likely the case. First off, the party was to celebrate Bruce Wayne's birthday, and Bruce doesn't like attending parties (most especially those where he's the spotlight, unless it serves another purpose/he has an "agenda"). Secondly, Bruce just doesn't like the rich and powerful of Gotham. Generally speaking, he finds them fake and pompous.

I mean by doing this he ruined the Wayne name as one person even said "The apple really has fallen far from the tree Mr. Wayne".

For Bruce, although his comments did significantly tarnish the Wayne name (i.e., his father's name), once he found out that Ra's Al Ghul was there, perhaps he considered having to kick them out as a kind of "two birds with one stone" thing. I could even see him getting a small amount of satisfaction from it too, though, in a light and jokingly way (again, because he doesn't really care about those people; just Rachel, Alfred, and now Lucius).

As Bruce has made very clear, Rachel is extremely important to him; more important than his family name. You can see though, after he makes those comments, that he expresses a degree of remorse, but that soon changes once he takes notice to Ra's Al Ghul's posse.

The other answers cover the main reason for the rude and offensive behavior: he wanted everybody to leave as soon as possible.

But, as noted in the comments, this does terrible damage to his reputation and the Wayne family name.

If he was going to lie about himself to get rid of the guests, he could have just as easily lied about the situation:

I'm sorry everybody, we have just received a bomb threat. Everybody please exit quickly.

This spares the reputation and family name and likely gets people to leave even faster.

So, considering that Bruce's behavior was not the most elegant or efficient way to achieve the goal, and Bruce is known to be a very elegant and smart man, another answer to the question:

Why was Bruce Wayne so harsh/rude when he asked for everyone to leave his party?

... is simply that the scene was exaggerated for dramatic effect and/or to showcase Christian Bale's acting skills.