It's not appropriate for the Executive branch to make requests to remove content that is political for political reasons. This has a chilling effect[1] and can thus be rightly judged by the populace as being against our shared reckoning on free speech.
I don't agree with the request, but I can understand where he's coming from. Obama is personally responsible for the lives of every government employee working overseas. Directly or indirectly, it is his orders that put them there. When the protests started, I'm sure it crossed his mind that somewhere, some of his people were going to die in retaliation for the movie. If I was in his shoes, and I thought that the voluntary removal of the video might save some American lives, I would have asked too. I mean seriously, a phone call or an email to save a person's life.
Unfortunately, there is a fundamental flaw with his reasoning. The extremists who actually conduct such attacks have no intention of backing down if their demands are met, so once the video was out, someone was going to die anyways.
They made a public request whose intent is at least partly to persuade people in the Middle East that the American government does not support the bigoted views of the filmmaker. The government has an entirely legitimate wish to dissociate itself from this film and dispel the perception that it is intended as a piece of propaganda.
They also asked the pastor not to do the Koran burning. If there is no implied legal threat, what is the big deal? The difference in this case is going after a third party host rather than the original poster.
They could have done lots of things. Using soft pressure in various ways is not something that's new to governments or generally detrimental or something that's ever going to go away.
I guess Obama could also have written an anonymous letter to the editor instead of having a speech announcing and advocating his jobs programs. But that's not how the real world works. Obama openly and strongly advocating X, Y and Z is not tyranny.
I would say this is more of a security issue. If the protests had been peaceful, and had there been no deaths, I doubt this request would have ever been made.
The mob was the specific instrument in this instance. Mobs are used regularly for similar purposes.
IMO the question at hand is whether as soon as some group threatens violence the USA kowtow. This appears to be the case here - the USA administration intervening in the free speech of a citizen in order to further the aims of a group because that group has threatened (and acted) violently.
1: http://en.wikipedia.org/wiki/Chilling_effect_%28law%29