It is always amusing to me when people talk about Romney and the issue with the church and holding office and they ignore Harry Reid et al. But there is an even BETTER example. That is the Secretary of Agriculture under President Eisenhower. His name was Ezra Taft Benson, and not only was he LDS, but he was a member of the Quorum of Twelve Apostles, the ruling body of the Church! During the eight years he served as the Secretary of Agriculture, he continued with some (if not all) of the duties associated with his church calling.
If there was ever a person who would have exemplified a conflict between the LDS religion and political necessity, Benson would have been it. But by all accounts, he did a reasonable job as Secretary of Agriculture, and was actually one of only a few of the members of the Eisenhower cabinet to stay with the President through both terms.
Just another data point, that is a bit older than some of the others, so may not be as well known.