The Great Escape is based on a historical event that was considered heroic and inspirational. Therefore, it is correctly classified as a drama.

Life is sometimes full of funny moments and The Great Escape certainly demonstrates that even in the direst circumstances, humanity's sense of humor is one of it's best defenses against despair. However, for the most part, the humor that you see in the movie is either secondary or ironic.

Nicholas J. Cull, professor of American Studies at the University of Leicester, specialist in film, media, and propaganda history, wrote an article “Great Escapes: ‘Englishness’ and the Prisoner of War Genre” (academic, peer-reviewed, published in Film History: An International Journal, volume 14, pp 282-295 (2002), obtained via my university library). In it he explores evolution of genre of POW films, and its relation to the concept of “Englishness” which he defines as much a feature of English national identity, as it is an ideological construct, “English exceptionalism” so to speak.

Origins of the British POW genre

Author explains how “Englishness” in context of WW II, despite the real historical complexities, became a simplified tale of resistance to tyranny where “at its heart is an emphasis on an island standing alone and the sense of community generated by a shared purpose”.

The overlap between the traditional expectations of Englishness and the English version of the Second World War help to account for the national attachment to the story. From the war to the present there have been a number of stories that have been told and retold about the English at war. […] Of these maybe the most explicitly English is the prisoner of war genre. If the received version of the Second World War is the microcosm of English myth, the Prisoner of War genre is the microcosm of the Second World War.

Thus Cull indicates that although “the genre was all the more powerful for its claim to authenticity” (because of rather close following to actual experiences based on memoirs and involvement of former POWs in film production) it was still formulaic and familiar from story to story with ever present English ingenuity in improvisation (“muddling through”), and even “perky defiance”. It was so familiar, in fact, that by the mid-50s contemporary critics already complained. Author continues: “By the 1960s it had become possible to joke about the POW experience”. In Very Important Person (1961) “the cliches of the genre were thoroughly rehearsed to comic effect”.

In the important section “The Appeal of the Genre” Cull explains why WW II POW stories enjoyed such success. By setting the action in the camp war was condensed to the face-to-face confrontation of archetypal “good guys” and “bad guys” characters. Issues of gender and class could be avoided. While women equally participated in the war effort, there was hardly a place for them in the POW film with rare exceptions. The fate of enlisted working class men who did not have “the responsibility to escape” as officers did, and simultaneously could be subjected to hard labour was not explored until The Password is Courage in 1962. That film clumsily touched up on the topic of the Holocaust and war atrocities in general — also the first in the genre. By populating the camps with British officers, it also gave the sense of the unique British worth to the larger victory, while avoiding the fact that Soviets did most of the fighting, and Americans bankrolled the Allied effort. Author puts it eloquently:

Set the film early in the war or in an RAF only camp and there is no need to discuss the American presence at all. The POW movie became a perfect tunnel through which the British could escape from the Americans.

Hollywood take on the genre in the Great Escape

Notable POW films outside the British formula were Hollywood’s Bridge on the River Kwai, and 1962 Le Caporal Epingle (The Vanishing Corporal) by Jean Renoir (author of 1937 La Grande Illusion incidentally depicting the POW life in WW I): films “marked by a focus on character and motivation quite alien to the tales of escape from German camps” of the British genre. Yet it was Hollywood’s big-time production of the Great Escape that brought the British story to the genre’s climax with its American treatment.

At the insistence of the veteran Hollywood screenwriter W.R. Burnett who “was appalled by the lack of American characters” were created “the wise cracking characters later played by Steve McQueen and James Garner”. “The final version was a compromise between the British and American perspectives penned by the Australian writer James Clavell.” While the historical reality was quite different:

The final film elided many significant details and characters from the actual events documented in Brickhill’s book. Although Americans were involved in the early stages of the historical events depicted by Brickhill, American POWs were actually moved to an adjoining camp before the escape began.

Introduction of American characters was not done only for the sake of such, but also to provide the British v. American dynamic typical for the time:

The use of British characters to symbolise “lovers of war” and Americans as reluctant but resourceful cynics was typical of the Hollywood representations of Britain at War in this era.

Author elaborates:

The film’s sub plot develops in ways that not only marginalise English characters in their own story, but makes their behaviour explicitly “other”. The American characters are the norm, and their attachment to things like baseball and improvised celebration of 4 July are the most obvious expressions of national culture in the film, to differentiate them from the Englishness of the camp. […] There are many moments in the film in which it becomes clear that we are watching an American depiction of Englishness.

The British-engineered plan becomes futile exercise for which “participants pay a heavy price”.

…the wisdom of the plan is called into question in the final moments of the film. The down-beat ending reflected the growing willingness to engage with the scale of Nazi atrocities, and provided a reminder of the grim reality underpinning the story of the escape.

Conclusion

The above lengthy summary serves to drive a point: while The Great Escape injected cultural confrontation and its down-beat finale provided for questioning of the whole concept, it is ultimately a derivative of “the classic up-beat prisoner of war story”. The story decidedly up-beat, in which aforementioned “perky defiance” is an important feature, not an afterthought nor coincidence.

I think, categorizing The Great Escape as “historical drama” can only be done at a stretch, as it is not quite historical (though indeed based on the true story), and not quite serious and dramatic. I would definitely not put it in the same row as, say, recent Hart’s War when it comes to dramatism or seriousness of the action.

According to Cull, animated Chicken Run (2000) is not only the homage to the Great Escape and the British POW film in general, but in some ways a profound satire of the genre.

By retelling the story with chickens Park and Lord’s film affectionately paid tribute to the POW genre but at some level drew a line under the stories as products of the past.

Chicken Run puts the chickens in the real peril, simultaneously alluding to atrocities of war, and placing the characters under the threat of chopping block:

The film also hints at the darkest aspect of the war. The chickens discover that they face not just captivity but industrial mass extermination.

This elevates the conflict from some “sporting thrill” for the officers, as it had only been acknowledged within the genre in 1981’s Escape to Victory by Michael Caine’s working-class character who exclaimed: “Your escaping is just some bloody upper-crust game!”

I don't think it's really valid to compare Hogan's Heroes to The Great Escape. Hogan's Heroes was definitely a comedy and intentionally omitted the more gruesome aspects of being a prisoner of war. For example, even though Hogan's men blew up bridges and trains and such, they never showed a person being killed or seriously injured. Also, in Hogan's Heroes most of the Germans are portrayed as being totally idiotic and easily manipulated and were not feared by the prisoners at all.

Contrast that to The Great Escape, where we see graphic scenes of prisoners being killed, like when the mole attempted to climb the fence or the ending when 50 escapees were executed.

There are certainly moments of humor in The Great Escape, but overall it is correct to classify it as a drama. As Mistu4u pointed out, the movie was based on a book, and the events in the movie actually took place (although of course some of the details were fictionalized or exaggerated). In fact, some of the Nuremberg trials after the war centered on who ordered the execution of the fifty men, since that was considered a war crime. During the trial of Herman Goering, he claimed that Hitler himself ordered the fifty to be executed (see this link for more info)

The movie is based on the book of the same name. The book is:

.... an insider's account by Paul Brickhill of the 1944 mass escape from the German prisoner of war camp Stalag Luft III for British and Commonwealth airmen. As a prisoner in the camp, he participated in the escape plan but was debarred from the actual escape 'along with three or four others on grounds of claustrophobia.

It does not look like a satire at all. Although it is true that some scenes here are portrayed as to show funny, but in the end it isn't. We should also keep in mind when to say a film is parody and when not. Here I found a good note:

Parody is a comic technique that imitates a previous text for the purposes of ridicule. For instance, in the film The Great Escape (1963) the character played by Steve McQueen is repeatedly thrown into solitary confinement ("the cooler") where he bounces a baseball against the wall to pass the time until his release. In the parody film Chicken Run (2000) the chicken Ginger gets sentenced to solitary confinement in a coal bin and bounces a rock against one wall to pass the time. The camera angle, the character's posture, and the sound of the ball bouncing off the wall all replicate the familiar scenes in The Great Escape . In order for this moment to function as parody for the audience, the spectator must be aware of the cinematic precedent, and able to connect it to the imitation (for the many young children who enjoyed Chicken Run , a coal bin is just a coal bin). There also must be a twist or element of comic difference to the imitation—in this case, the fact that the prisoner is a chicken and not a soldier.

So taking all these issues into account, we can say it was not meant to be shown as a satire. However we know there are a plenty of films which were created thinking of a genre, but might result in more successful as movie of another genre. The whole thing is about personal choice. The genre is decided by the majority of viewers. However in this case, the majority has chosen history and that is it!