If you are instructed to lay more eggs than you have available spaces on your birds, any excess eggs beyond your birds' egg limits are lost and not placed.

This is supported by the following rulebook references:

• "A bird card can never hold more than this number of eggs."

• "It is possible that you will have more capacity to lay eggs (based on your player mat) than you have spaces on your birds. Any excess beyond your egg limit is lost."
  (Page 8)

• "If you attempt to lay more eggs than your birds have space for, the extra eggs are lost."
  (Page 1)

Summary:
You cannot place more eggs than your birds' combined egg limits allow; any extra eggs are lost.

Sources:
Page 8, Page 1

Source: Core Rulebook (pp. 1, 5, 8, 9)