A 2-1 defeat to Japan, 1-1 draw with Spain and a 4-2 victory over Costa Rica were not enough to secure passage to the round of 16.
This is the second straight edition of FIFA's flagship tournament the 2014 winners have failed to make it out of their group.
As defending champions in Russia four years ago, Germany under Joachim Low, finished bottom of their group, which comprised Sweden, South Korea and Mexico.
This time in Qatar, led by Flick, Germany could only secure third spot behind Japan and Spain.
Germany are due to host EURO 2024 and Flick's contract runs until the end of the tournament but Hamann does not believe the former Bayern Munich boss should be the one to lead them on home soil.
Speaking to Sky Sports in Germany, a furious Hamann said: "I think it's impossible that we can continue with the coach.
"After this debacle, we only have 18 months until the European Championships at home, that was pathetic.
"Costa Rica had one shot on target in the first two games and we made them look like Brazil. In the team, everyone does what they want. I see no reason why Flick should remain as coach.
"When I hear some comments from those responsible [with other teams], I have to say that responsibility looks different. Excuses, excuses, excuses. It's time for a cut."
