As Clint already mentioned, the obligation to bring doves applies to a Zava - one who bleeds between the expected times of her period (to oversimplify). So most women never had this obligation.
Another missing piece is that the woman does not have to bring the doves immediately - she can accumulate the obligations and bring them all together.
As long as she has not brought them she is a Mechusar Kapara - missing atonement - and she may not eat Korbanot, as the Rambam codifies in הלכות מחוסרי כפרה פרק א. Not bringing your birds has no other ramification.
The only place you can eat Korbanot is the same place you can bring your doves - or pigeons.
Practically speaking, this meant that when they went to Jerusalem for the 3 festivals, she would bring her birds so that she could then partake of the festive sacrifices.
By the way, the others in this category of מחוסרי כפרה are: a Zav, a mother after childbirth, a Metzora and a convert.