Why do heroes in classical greek tragedies often end up suffering?

The term tragedy refers to literature that encompasses the downfall of the protagonist whether it be by the interference of other characters or because of their own character flaw. So in any tragedy, the protagonist is meant to suffer or reach their downfall because that is the exact definition of the term

Hope this helps

There is a reason it is called a "tragedy" instead of a story. Yet, as for other reasons, one of the main ones is their fatal flaw. The great heroes of old Greek literature were meant to be built up, do great things, and then fall horribly, encouraging the ancients because it made them feel better, knowing that heroes did so poorly as well. The hero always had an extreme fatal flaw (anger [Hercules], lack of focus [Theseus], loyalty [Jason]) that would lead to them making a choice that would secure their death and destruction, if not worse. Hope this helped.