Klinsmann, a World Cup winner with West Germany in 1990, claimed while working as a pundit for the BBC that Queiroz's side had "worked the referee" in their 2-0 victory over Wales on Friday.
The former Tottenham forward said: "Carlos fits really well with the national team and their culture, he failed in South America with Colombia and then failed to qualify with Egypt, and he came in right before the World Cup with Iran, where he worked for a long time.
"It is not by coincidence, it is part of their culture, how they play.
"They worked the referee. They work the linesman and fourth official, they are constantly in their ear. There were a lot of incidents we didn't see. This is their culture, they take you off your game."
Those comments clearly enraged Queiroz, who took to Twitter on Saturday to respond.
"No matter how much I can respect what you did inside the pitch, those remarks about Iranian culture, (the) Iran national team and my players are a disgrace to football," he wrote. "Nobody can hurt our integrity if it is not at our level, of course."
Queiroz went on to invite Klinsmann to the team's training camp and meet the players "to learn from them about the country" before concluding: "We just want to follow with full attention what will be the decision of FIFA regarding your position as a member of Qatar 2022 Technical Study Group.
"Because, obviously, we expect you to resign before you visit our camp."
The Technical Study Group is led by former Arsenal manager Arsene Wenger, who is FIFA's chief of global football development, and provides analysis on the World Cup action to identify styles and trends which develop.
It is not Queiroz's first outburst at the tournament.
He criticised Iran fans for not supporting the team in their opening defeat to England, and then confronted a journalist at a press conference on the eve of the Wales match, wanting to know why England manager Gareth Southgate does not face the same level of questioning on political matters as him.