The area marked in the map is of course the Mitte, the old, presigious central part of the city including such sights as the Brandenburg Gate, the Museuminsel and the Alexanderplatz, etc.
I guess with whatever East-West crises, the party that is occupying the Mitte at the time the crisis breaks out would be highly unwilling to give up the zone to the next in line, lest the other side stops respecting the deal after they get in... Especially as it is a location with such iconic sights. It would be an obvious site of protests - for example I could see the USSR drumming up large East German communist demonstrations in the area every time it is their turn to lord over it. And so it would be an obvious flashpoint, some might say more trouble than it is worth.
I think to get a realistic idea how a crisis would affect such an area, you might do a play-by-play of how the different sides would react, for example then the 1948 crisis started, assuming no major changes caused by that time. Taking control of the Mitte would be one of the Soviet forces' first actions in this case, I would imagine, and who's contractual turn it is to hold it would have an effect on how the situation develops.
Looking somewhat further into the future, if the Berlin airlift and following events would not cause the USSR to claim the Mitte all to its own, and if the turn-based system can somehow be kept going into the 60s and the 70s, say, then the existence of the zone would have a lot of bigger and smaller knock-on effects. The Wall of course would have to be built behind it, and this would change public transport and the division of the metro lines, say. It would also change local architecture, for example. IOTL the area become something of a GDR playground, where the Palast of the Republic and the Berlin TV tower were built as examples of East German strength and modernity. ITTL, this would not happen as the area would have no permanent owner. It might be difficult to negotiate over the upkeep of the old buildings and infrastructure, as well as building new things, when the area would not be a part of either German state (presumably). It could lead to either the area starting to fall apart, or then some pretty exotic renewal projects (like a full redesign and rebuild of the Mitte by the neutral Swedes and Finns, working under a UN mandate, in the late 70s).
And there's a lot more in terms of butterflies, I believe. All in all, "The Fourty-Four Occupations - the Unlikely Story of Berlin Mitte, 1945-1989" might have the makings of some pretty quintessentially twisted Cold War black comedy...