To get a better idea of whether this was possible I looked at the weight and balance for a 737. The Airplane Characteristics document provides the necessary information to estimate how the center of gravity (CG) may shift due to passengers deplaning.
The information in the document shows between 89% and 96% of the weight on the main landing gear when the aircraft is within the proper CG range for a 737-800. The max landing weight is about 66,000 kg and the max zero fuel weight is about 62,000 kg, so a weight of 60,000 kg when arriving at the gate seems reasonable for a full flight. If 94% of the weight is on the main gear, this puts the CG at about 1m forward of the main gear, which is towards the aft end but within limits.
I considered the high density layout shown in the document, which is a single-class layout with 175 economy class seats at 32 inch (0.8m) pitch. The first row has just 3 seats and is located about 14m forward of the main gear, with each row of 6 located another 0.8m aft.
I assumed that the aircraft is unloaded from the front, one row at a time, and that each passenger (plus carry-on) weighs 85kg. For a rough idea of the results, consider that the CG starts with a moment of about 60,000 kgm1, and the rows forward of the main gear are an average of 7m forward, so removing about 8600kg from those rows will move the CG aft of the main gear and tip the plane. This means about 100 passengers, which is within possibility for 18 rows of seats.
So with a 60,000 kg aircraft, 94% of the weight on the main gear, and 85kg passengers, the plane will tip after row 13 disembarks. This is very sensitive to CG. If the CG starts at 93% of weight on the main gear, the plane won't tip. If the CG starts fully aft (96% weight on main gear), the plane tips at row 7.
However, the plane isn't really unloaded 1 row at a time. Usually there is a line of passengers down the aisle. If there is 1 passenger per row in the aisle, the plane won't tip at 94% weight on main gear, and won't tip until row 11 at 96% weight on main gear.
A heavier plane will be harder to tip, and heavier passengers will make it easier to tip. Planes that have upper class passengers in the front will be harder to tip, since less weight is removed from the front of the plane. A longer plane (737-900) will actually be slightly harder to tip, as will a shorter aircraft (737-700)2.
So why doesn't this ever happen?
As I mentioned, it's very sensitive to CG. Only an aft CG will be an issue. Also, passengers in the aisle will help balance things out. Many airlines have larger seats with more room in front that will reduce the effect of passengers leaving these rows.
Another factor is baggage. As soon as the plane stops and shuts down the engines, the ground crew will often start unloading the plane. It will take a bit longer to position the jet bridge or stairs and begin disembarking passengers. Removing weight from the rear will help balance passengers leaving the front.
In reference to the post that created this question, it's unlikely that one passenger would make the difference in tipping the aircraft. However, it is entirely possible that fully unloading the seats in front and not the back, and having no one coming down the aisle, is a risk factor that flight attendants would want to avoid.
1: I know that the typical unit is Nm but we only care about balance here, not forces, so there is no need to complicate things with gravity.
2: This may seem strange but for the same weight % (which is how I'm comparing) each aircraft will have a different CG location relative to the main gear, due to the distance between the nose and mains changing. Also, different aircraft have different weights and different CG ranges.