Zimbabwe, boosted by a surprise win over Australia, needed only a draw to reach the Super 8s, while Ireland required a victory to stay alive.
Persistent rain ensured the match was called off, with both teams sharing a point.
Zimbabwe move to five points and are assured of progressing, while Ireland finish their campaign on three.
Australia, with two points and one game left against Oman, are also eliminated. Zimbabwe will face Sri Lanka in Colombo to decide the Group B winners.
