In 1972, Linda Jenness
ran for President of the United States, although she was 31 at the time.
Many youth rights groups view current age of candidacy requirements as unjustified age discrimination. Occasionally people who are younger than the minimum age will run for an office in protest of the requirement or because they don't know that the requirement exists. On extremely rare occasions, young people have been elected to offices they do not qualify for and have been deemed ineligible to assume the office.
In 1934, Rush Holt of West Virginia was elected to the Senate of the United States at the age of 29. Since the U.S. Constitution requires senators to be at least 30, Holt was forced to wait until his 30th birthday, six months after the start of the session, before being sworn in.
In 1954, Richard Fulton won election to the Tennessee Senate. Shortly after being sworn in, Fulton was ousted from office because he was 27 years old at the time. The Tennessee State Constitution required that senators be at least 30. Rather than hold a new election, the previous incumbent, Clifford Allen, was allowed to resume his office for another term. Fulton went on to win the next State Senate election in 1956 and was later elected to the US House of Representatives where he served for 10 years.
In 1964 Congressman Jed Johnson, Jr. of Oklahoma was elected to the 89th Congress in the 1964 election while still aged 24 years. However, he became eligible for the House after turning 25 on his birthday, December 27, 1964, 7 days before his swearing in, making him the youngest legally elected and seated member of the United States Congress. 
In South Carolina, two Senators aged 24 were elected, but were too young according to the State Constitution: Mike Laughlin in 1969 and Bryan Dorn (later a US Congressman) in 1941. They were seated anyway.
On several occasions, the Socialist Workers Party (USA) has nominated candidates too young to qualify for the offices they were running for. In 1972, Linda Jenness ran as the SWP presidential candidate, although she was 31 at the time. Since the U.S. Constitution requires that the President and Vice President be at least 35 years old, Jenness was not able to receive ballot access in several states in which she otherwise qualified. Despite this handicap, Jenness still received 83,380 votes. In 2004, the SWP nominated Arrin Hawkins as the party's vice-presidential candidate, although she was 28 at the time. Hawkins was also unable to receive ballot access in several states due to her age.