The exhibition, launched on Monday, shows 204 entries from Iran and abroad.
Masoud Shojai, head of the country's "Iran Cartoon" association and the fair organiser, said that "we staged this fair to explore the limits of freedom Westerners believe in".
He said: "They can freely write anything they like about our prophet, but if one raises doubts about the Holocaust he is either fined or sent to prison."
At the opening ceremony of the month-long fair in Tehran's Palestine contemporary art museum, Shojai said: "Though we do not deny that fact that Jews were killed in the [second world] war, why should the Palestinians pay for it?"
He added that around 1,100 cartoons were submitted by participants from more than 60 countries and that more than 200 are on show.
One cartoon by Indonesian Tony Thomdean shows the statue of liberty holding a book on the Holocaust in its left hand and giving a Nazi-style salute with the other.
Shojai said the top three cartoons will be announced on September 2, with the winners being awarded prizes of 12,000, 8,000 and 5,000 dollars respectively.
Muslims angered by the Danish
cartoons protested worldwide
He did not elaborate on the source of the prize money, but emphasised that it did not come from any governmental body.
The fair is being staged by Iran Cartoon and the country's largest selling newspaper Hamshahri, which is published by Tehran's conservative municipality.
The contest was announced in February after caricatures of the Prophet Muhammad were first printed in Denmark and then picked up and published worldwide, enraging Muslims.
Many Muslims considered the cartoon offensive and a violation of traditions prohibiting images of the prophet.
The entries on display came from nations including the United States, Indonesia and Turkey.
About 50 people attended the exhibition's opening.
Zahra Amoli said: "I came to learn more about the roots of the Holocaust and the basis of Israel's emergence."
Iran's fiercely anti-Israeli regime is supportive of so-called Holocaust revisionists, who maintain that the systematic slaughter by the Nazis of mainland Europe's Jews and other groups during World War II was either invented or exaggerated.
Mahmoud Ahmadinejad, Iran's president, has also prompted international anger by dismissing the Holocaust as a "myth" used to justify the creation of Israel.