First-round competition was scheduled for March 24-25.
"The health and safety of players, coaches, staff and fans is our main priority," U.S. Open Cup commissioner Paul Marstaller said in a statement. "Given the unique nature of the competition, which encompasses clubs from multiple leagues that have suspended their seasons, we are taking this action and will look to determine future dates to allow the 2020 Lamar Hunt U.S. Open Cup to be played."
Teams from Major League Soccer, USL Championship, USL League One, National Independent Soccer Association and various amateur organizations compete in the tournament. MLS suspended play for 30 days earlier this week, and other leagues took similar action as coronavirus concerns increased.
Second-round action was slated for April 7-9 and third-round play was scheduled for April 21-23.
The U.S. Open Cup began in 1914 and has been played every year since its inception.
The Atlanta United won the 2019 championship, defeating fellow MLS side Minnesota United 2-1 in the final.