(As for the allegedly greater accuracy of pinyin; it's true that "Beijing" is closer to the Mandarin pronunciation than "Peking," but they're equally far from, say, Cantonese "Pak-king," which is just as Chinese.)
All right, I could partially understand the desire not to offend the authorities in China, a large and increasingly important country with which we want to do business. But what happened when the repellent junta that rules Burma decided they wanted us to say Myanmar instead of Burma and Yangon instead of Rangoon? Exactly the same thing. Nobody replied "Fie on you, vile dictators! Aung San Suu Kyi and other democrats prefer 'Burma' and say they will restore the name when they can, so we laugh at your pretensions!" No, they hastened to change all the books again, and complain when people used the unreconstructed terminology. When I made a comment to that effect on MetaFilter, two people took me to task for my cultural imperialism. I'm sure they aren't junta supporters, yet they automatically took the side of the powers that be. And the same thing is going on now with India; the extremist Hindus running the place are insisting on "Mumbai" for Bombay and "Kolkata" for Calcutta, and they seem to be getting their way. [Note: For a better-informed corrective to this hasty sentence, see Kaushik's comment (#12).] Prepare to start saying "Bharat" for India soon.
Now, here's what I don't get. Nobody seems to mind that Spanish-speakers say Nueva York, that the French refer to la Nouvelle-Orléans, that the Chinese call this country Mei Kuo—excuse me, Meiguo—and the Russians Soedinyonnye Shtaty Ameriki. As far as I know, nobody cares that the French refer to Regensburg as Ratisbon or that the Hungarians call Paris Parizs. So why this concern for the English names of foreign places? And why in the name of Babel does this country, in every other way so self-satisfied and downright imperial, jump when even the pettiest dictator says "froggie"? All suggestions will be much appreciated. (And hell, feel free to accuse me of cultural imperialism if you like, just so you answer the question.)
Addendum: Scribbler brings up an excellent point in the comments: isn't the name of a country determined by the government of that country? He gives as his example Burkina Faso, formerly Upper Volta. Now, that's an interesting example, because according to my bible in these matters, Pospelov's Geograficheskie nazvaniya mira, the name was changed precisely because the old one had so many variations: Upper Volta, Haute Volta, Alto Volta, Verkhnyaya Vol'ta, etc. This was seen as impractical, and the new name (which apparently means 'land of upright/honest/incorruptible people') was meant to provide a unique designation that would remain the same cross-linguistically. This can be called the poster boy of country-name change; it completely changes the name (for a sensible reason, even), and there seems no prospect of the name changing back. Burma/Myanmar is the opposite: the two are alternate modernizations of the same Sanskrit preform, so that in some sense they are "the same name," and there is every prospect of the preferred English version returning to Burma (if, as we all hope, the current thugs are tossed out). Another point is that there was not much occasion to refer to Upper Volta, so that the name change didn't cause many problems; Burma is much worse from that point of view, and the Chinese situation still worse. Obviously each case must be evaluated on its merits, but my point still remains: English-speakers have a right to their traditional place names (and pronunciations: Lyons used to be pronounced "lions" and Milan "MY-lun," but they changed in the natural course of events, not by diktat from abroad or above).
Added addendum: Renee has responded with a touching and poetic entry in her own blog, on the onomastic history of her hometown Lwow/Lviv/Lemberg and the ghosts of buildings; I urge everyone to go there at once.