I couldn't find a picture of Truman with Mickey, but here's Truman at the Disneyland. By the way, Truman was the first president to visit Disneyland.
Truman at Disneyland, 1957
Conan O'Brien talks about Truman and Disneyland. video link
Again, no picture of him with Mickey, but here's Eisenhower at the Disneyland Fire Station.
Eisenhower at Disneyland, 1961. credit
No picture of Johnny with Mickey, but Kennedy visited Disneyland while he was Senator. Following the assassination of Kennedy, Disneyland closed it's doors the following day to mourn his death. His son John Kennedy Jr met Mickey.
Kennedy at Disneyland, 1959. credit
Disneyland closed to mourn Kennedy's death
Kennedy Jr meeting Mickey at the World's Fair, 1964. credit
Though he never met Mickey or visited Disneyland, Johnson awarded Walt Disney with the Medal of Freedom in 1964.
LBJ and Walt Disney, 1964. credit
No picture of Nixon with Mickey, but Nixon as Vice President opened the Disneyland Monorail along side Walt Disney. Also his famous line "I'm not a crook" was said in a press conference held at the Walt Disney World Contemporary Resort.
The Nixon Family and Walt Disney opening the monorail, 1959. credit
Video of the ball room where Nixon said "I'm not a crook." video link
Ford has met Mickey Mouse and celebrated Mickey's 50th Birthday.
Gerald Ford and Mickey, 1978. source
Carter has met Mickey Mouse and held a birthday party for him at the White House in 1978.
Jimmy and Mickey at the White House, 1978. credit
Reagan has met Mickey and also he was one of the special guest reporters at the opening of Disneyland.
Ronald and Nancy at Disney World. credit
Ronald at Disneyland opening day, 1955. credit
Bush Senior met Mickey
Bush Senior at Disney World. credit
I didn't find a picture of Bill with Mickey, but I found a picture of the Clinton family with Donald Duck from 1984.
The Clinton Family with Donald, 1984
Hillary with Mickey at Epcot, 1993. credit
No image found of him and Mickey. I'm pretty sure that he has visited a Disney park, but I have not found a picture of him at the parks. Though I found a picture of him holding a Disney World autograph book.
No picture of Obama and Mickey, but Obama has visited the Magic Kingdom in Disney World and found a picture of him holding a Mickey hat that reads "Senator Obama." Also here's Michelle with Bob Iger and Mickey Mouse.
Senator Obama and his Mickey ears. credit
Obama at Magic Kingdom, 2012. credit
Bob Iger, Mickey and Michelle
Whether you like him or not, he once hosted a party for Disney World's 20th Anniversary.
Minnie giving Trump a Mickey watch, 1992. credit
Trump with Snow White and the Sevens Dwarfs, 1994. credit
As far as I know, The Royal Family or the Windsors has not meet a Disney costumed character or visited a Disney Park. But here's Princess Diana meeting Roger Rabbit.
Probably the first time a member of the royal family has touched a Disney costumed character, 1988