What does opening the door for Elijah symbolize?

I am assuming your question is referring to the Jewish Passover celebration in which participants symbolically open the door, welcoming the prophet Elijah to enter the home.  An extra place at the table is also set in anticipation for the guest.  These symbolic actions concerning the anticipation of the prophet Elijah are done every year at the Jewish Passover.

This expectant and hopeful anticipation of the prophet Elijah hearkens back to a prophecy in the Jewish book of Malachi, found in the Old Testament of the Christian Bible.  In Malachi, the God of the Hebrews says, "Behold, I send My messenger, and he will prepare the way before Me" (Malachi 3:1, NKJV).  The Jews understood this to be Elijah, who would appear to herald the coming of the Jewish Messiah.  Jews, anticipating the coming of Messiah for thousands of years, still today anticipate the prophet Elijah, who announces the coming of Messiah, by setting a place at the table and opening the door for his entrance.