It's symbolic, because it's a door into the Phanar, the headquarters of the Ecumenical Patriarchate there, which was welded shut after the Turks hanged Patriarch Gregory V after the 1821 Greek rebellion.
Now, in an effort to prove that Turkey has religious freedom and merits entry into the European Union, the Turkish government is thinking of allowing the Orthodox Seminary at Halki, closed since 1971, to reopen. The Greek community has been progressively hemmed and harried from the Ottoman conquest until now. Mattingly writes:
There are fewer than 2,000 Greek Orthodox Christians left in Istanbul, most of them elderly. Turkish law requires the patriarch to be Turkish and, quite literally, if the current patriarch died tomorrow he would be almost impossible to replace.
He adds that the Turkish government says it will reopen the seminary when the Church opens the welded door. The Church replies that it will reopen the welded door when the Turkish government permits it to open the seminary. It sounds like a squabble going on right now between two high-school girls of my acquaintance, but I guess that shows how important the symbolic gestures are rather than how trivial the parties are.
I wonder if it's too late to save the Patriarchate of Constantinople, one of the five great patriarchates of the ancient Church.