From a CQ Politics FAQ about the conviction of Sen. Ted Stevens on seven counts of of filing false financial disclosure forms:
Q: Did Stevens have to give up any perks of office?
A: Yes. Senate Republican Conference rules require a member indicted on felony charges to step down from the post of committee chairman or ranking Republican on all committees and subcommittees on which they serve. Long before his trial, Stevens, the Senate’s longest-serving GOP member, relinquished his role as the top Republican on the Defense Appropriations Subcommittee, the Commerce, Science and Transportation Committee and the Homeland Security and Governmental Affairs Subcommittee on Disaster Recovery within hours of his indictment.
Prior to the Democrats taking control of the Senate in 2007, Sen. Stevens was President pro tempore of the Senate, putting him third in the line of presidential succession. So my question is this: if Stevens had still been serving as President Pro Tempore when he was indicted, would the Conference rule cited above have required him to step down from that position? It's not really a leadership position; it's just a title that is automatically granted to the most senior Senator from the majority party. On the other hand, it is a Constitutional office, and the person who holds the position is, to coin a phrase, three heartbeats from the Presidency, so it's not hard to imagine that he would have been required to give it up.
(For what it's worth, Stevens did retain the title President pro tempore emeritus even after being indicted; the title is purely honorary, but it does entitle him to extra staff.)
Another interesting (if you are a Congress nerd) question: assuming Stevens had been indicted while serving as President pro tempore, and had been forced by the Republican Conference to step down from the post, who would have taken over the post? Answer: Sen. Pete Domenici of New Mexico, who is the second most senior Republican Senator (and fifth overall).