Mr. Burns' age has fluctuated during the course of the show, being explicitly stated as 81 in "Simpson and Delilah" and as old as 104 in "Who Shot Mr. Burns? (Part One)", "Homer the Smithers" and "A Hunka Hunka Burns in Love". In "Fraudcast News", Burns claims that he was born in 1881, making him 123 years old when the episode aired in 2004. In an issue of Simpsons Comics, Mr. Smithers states Mr. Burns is 118 years old. Even he's 150 years old He has been referred to as "Springfield's Oldest Resident" with a Social Security number of 000-00-0002 (cursing Franklin Roosevelt for having 0001). The first Social Security cards were issued in 1936.
Frequent allusions to Mr. Burns' age place his date of birth in the late 19th century, which would make him more than 100 years old. He says he had a driver's license that expired in 1909, graduated from Yale University in 1914, and was depicted in a 19th-century woodcut terrorizing children (which, surprisingly, features him as an old man). Although Burns appears younger than Abe Simpson in all the World War II flashbacks, he is always older than Abe in all other sources. In "Bart Stops to Smell the Roosevelts", Bart sees a picture of Burns being punched by Theodore Roosevelt while Bart is researching him. In this instance he looks the same as he does today. Mr. Burns has also claimed that he has survived 12 economic recessions and eight panics. Other references to Mr. Burns' age place him at thousands of years old. In "Them, Robot", when Homer and Mr. Burns believe they are about to die, Homer asks Burns what his real age is. Although he doesn't give his age outright, he states, "Well, it's four digits"; if this is true then Mr. Burns is over a thousand years old. His ATM password, which is the same as his age, is four digits long (although it is possible the first and/or second digits are 0). Mr. Burns' birthplace has also been named as Pangaea.
Also, in the one-time appearance of his "league of evil", a group of evildoers from many eras of history, many of the members are quite ancient historically. Since Burns presumably knew all of them before they died, this pushes his age back at least another 500 years, if not much more. Burns' wine cellar also has trophy heads of a Dodo, a Sabertoothed Cat, and a Tyrannosaurus rex.
Burns' age provides the writers a character with whom they can express dated humor and references to popular culture before the 1950s. From time to time he takes up arms against someone he perceives as his enemy, the joke being his feebleness and the archaic nature of his chosen weapons (e.g., a morning star in his limousine, a blunderbuss, dueling pistols, etc.). Familiarity with the use of these weapons dates him considerably far back in history, especially the morning star, since this hasn't been a widely used weapon since the late medieval era.
However, because his mother, who is currently alive, is 122, Burns would have to be younger. If the age given in Simpson and Delilah is correct, she would have to have given birth to him at around 18 years old.