Politically, no, I don't think Harvard is materially more conservative/Republican in voting patterns or in, ultimately, the political beliefs of its faculty or student body compared to, say, Columbia, Yale or Princeton (though more conservative than those top schools with a distinctly liberal reputation, like Brown or UC Berkeley). Institutionally and structurally, I do think Harvard is somewhat more conservative than most, though less so than at least Princeton, but that's not the same as political conservatism.
The University of Chicago is the most politically conservative of the elite colleges (though it's still quite liberal on the whole). And then the University of Pennsylvania, although Penn is more for business-orientation rather than a politically conservative culture. After UChicago and Penn, I'd probably put MIT and the other engineering schools to the extent included (like CalTech) next, but it's a fairly big leap as only UChicago and to a lesser extent Penn has a politically relatively conservative reputation and a large conservative faculty presence.
(There's also the question of what an elite college is, but I think it's fair to include the Ivies, the near-Ivies, the would-have-been-Ivies, the elite liberal arts colleges and the top engineering schools. You're left with some corner cases like Johns Hopkins, Tufts, NYU, Duke, Georgia Tech and some of the most well-regarded state schools (UCLA, Michigan, Virginia -- not including Berkeley, which is clearly elite), but I don't think those really change things, and you still leave out schools like Pepperdine or George Mason that are actively conservative.)
Of the US News and World Report Top 20 in the college rankings, the most conservative is either Vanderbilt or Notre Dame. Neither are really conservative (I'm a Vandy alum and it definitely isn't), but they aren't
as liberal as the rest of the "elite" schools.
