What problems do you start run into when an aircraft is disproportionally long?
The question reminds me of the DC-8 Super 61. I always found its fuselage to be absurdly long (picture below, source).



Friction drag

First of all, your volume to surface ratio will suffer. A more slender fuselage will have proportionally more surface area for the same volume than a more stubby one. The first aerodynamic effect, thus, is a higher friction drag relative to fuselage capacity. Below a fineness ratio of five, increasing flow separation will unduly increase drag. Moving from an A340-style fuselage to something more slender, however, will not affect separation.

While volume would need to be held constant in case of airships, the goal for an airliner should be to maintain seating capacity, which can be assumed to be proportional to floor space (within limits - aisle height should not drop below 6 ft). Therefore, the figure of merit should be volume$^⅔$, and the growth in skin friction drag (and structural weight) should be less severe with increasing slenderness of the fuselage. Also, with a longer tail the tail surfaces can be made proportionally smaller, which reduces friction drag a bit, too.

The graph below is from S. Hoerner's Fluid Dynamic Drag, figure 24 of chapter 13, and actually valid for flying boat fuselages. It shows nicely how the drag coefficient grows with increasing fineness ratio; however, since the drag coefficient is based on frontal area, this is slightly misleading.



Hoerner's comments to this graph include an optimum based on volume$^⅔$:


  The optimum ratio with respect to drag on frontal area, seems to be in
  the vicinity of 5. Based on (volume)⅔, a minimum drag coefficient is
  obtained at l/"d" ≈ 9, corresponding to l/b ≈ 15 in the hull family
  tested


Maximum Cruise Speed

When being operated at speeds approaching Mach 1, the more slender fuselage will show a higher value of the drag divergence Mach number, so the second aerodynamic effect of a more slender aircraft is a slightly higher cruise speed, wing aerodynamics permitting.

Again, a graph from Hoerner's Fluid Dynamic Drag (figure 37 of chapter 15) as proof:



Pitch Damping

Third, since the fuselage length will determine the lever arm of the empennage, a more slender fuselage of the same capacity will exhibit a higher pitch damping relative to its static stability. Pitch damping depends on the square of the tail's lever arm while static stability grows only linearly. Note that the tail size needed for the same stability and pitch authority will change inversely to fuselage length, offsetting some of the increase in zero-lift drag of the longer fuselage.

Take-Off and Landing Speed

If the landing gear remains unchanged, a fourth effect could be a lower rotation angle on take-off and pitch attitude on landing in order to avoid a tail strike. This might result in higher take-off and landing speeds.

This answer would be much longer if aeroelastic and structural considerations would be included. I restricted it to aerodynamic factors as requested.