In at least two cases, members of the 91st (allegedly) interacted with Abraham Lincoln.
First, after Jesse Wharton was shot, the sentry, and the corporal who had ordered him to shoot, were arrested, and held in the Central Guard House. Weeks later, Major Todd, of the 91st, carried a petition to Abraham Lincoln. They were freed on the same day. (Thomas Walter, 'Personal recollections and experiences of an obscure soldier', Grand Army Scout and Soldiers' Mail, volume 3, number 35, page 2)
Second, in 1864, Hugh McLaughlin (H) went to the White House because a paymaster refused to pay him. According to the story (told decades later by William C Reiff, he pushed in past the guard, found Lincoln writing at a table, and made a case that he should be paid. Lincoln and his wife asked many questions about his service, and then ordered him to be paid. (William C Reiff, 'Lincoln's kindness', National Tribune 26 December 1895, page 3)
Lincoln saw Richard T McCarter (D) in the Armory Square Hospital, and arranged a furlough for him. ('Richard T McCarter', in The Pilgrims' book, containing the Articles of Constitution of the Pilgrims to the battlefields of the rebellion and other matters of interest to the members of that organization, compiled as of Memorial Day, 1911. Philadelphia: George H Buchanan Company, [no date]. Pages 73-81. available at http://babel.hathitrust.org/cgi/pt?view=image;size=100;id=mdp.39015064424495;page=root;seq=81;num=73 (accessed 17 October 2010))
Also, Joseph Burroughs (F 21st VRC) 'stood guard over the body of Lincoln after the assassination of the martyred president'. According to a family story, William Henry Harrison Beal was playing pool next to the theater when Lincoln was shot. And according to his death notice, Robert Sinex saw Lincoln assassinated, and participated in John Wilkes Booth's capture.
(See also Elections.)