Why did upward turning propellers in the F-82 Twin Mustang cause a loss of lift in the center wing section?
I'm not sure the history provided in the Wikipedia article is entirely accurate, for the fact that both engines in an outward spinning, two-prop counter-rotating configuration are critical.  This is confirmed in the introduction of this book, though it seems to be written well after the fact and without reference to the oft-quoted story of not getting enough lift on its maiden run:


  Turning the propellers toward the fuselage meant much better control during single-engine operation.


Likewise, the republished F-82 Operator's Manual boasts, "Single-engine control characteristics of this airplane are exceptionally good."  Again (but also as expected), no mention of any trial and error in the design phase.

And the more I researched the F-82's design, I found various and unsourced information.  Many sites repeated the initial flight test story, while another said the first attempt's failure was attributed to weight.  

My guess is that initial designs of the F-82 simply borrowed from the slightly earlier P-38's final configuration, where the outward-spinning counter-rotating propellers reduced downwash on the horizontal stabilizer and made for a more stable gunnery platform.

As for figuring out the lift cancellation in the F-82's center wing section, it is probably helpful to explain first what effect a single propeller has on the wing behind it.

Generally speaking, a spinning propeller creates a downstream vortex in the same direction as the propeller is rotating.  The area of wing behind the upswing of a propeller blade experiences a higher angle of attack to the resultant airflow, resulting in increased lift and drag across that section of the wing.


Image source: Civil Aviation Authority of New Zealand

Likewise, the opposite occurs on the downswing side, resulting in a net increase in lift and net decrease in drag behind the engine as a whole.  This single-engine airflow model is generally true for multiple-engine models with all propellers spinning in the same direction:


  The interaction of the slipstream with the wing produces an increase in overall lift coefficient. This net increase is the result of a local increase in local lift coefficient $C_L$ for the part of the wing located in the slipstream with up-rotation while the downrotation induces a local decrease in $C_L$. The local drag coefficient $C_d$ is locally increased in the uprotation part and decreased over the down-rotation section.  The net result is generally a reduction of drag on the wing.


So the F-82 seems to be an enigma in this regard.  Theoretically, an outward rotation should have increased lift on the center section, not reduced it.  So why didn't it?  Perhaps there was too much interaction between the two propeller wakes.  Again, consider the P-38 Lightning.  Though they both have twin counter-rotating engines and two tail booms connected by a shared horizontal stabilizer, the P-38 has a big center fuselage for the pilot sitting right between the two engines, effectively isolating the individual propeller slipstreams.

A study on the "Down Between Engines" (DBE) configuration of the Airbus A400M noted "[strong] interact[ion]" of its counter-rotating engines' slipstreams


  especially [in] the part of the wing located between both engines and results in a local decrease in lift coefficient that is not entirely offset by the local increase on those parts of the wings that experience upward motion (emphasis mine)


resulting in both decreased lift and increased drag in the wing section between the engines.  

Note that DBE (i.e. inward-spinning propellers) was the fix for the F-82's lift woes; that the inter-engine wing region still experiences decreased lift and increased drag in the more preferable counter-rotating configuration underscores the notion that interaction of propeller slipstreams is not desirable.  My guess is that this switch removed just enough of the adverse lift conditions so as to enable the aircraft to get airborne.  It would be interesting to find out if anyone modified an F-82 to remove the counter-rotating aspect.



UPDATE:
At long last, I have found what appears to be the most credible source for the Wikipedia article, a 2013 issue of Warbirds magazine.


The first flight was (at least somewhat) attributed to being overweight.  A second attempt on 16 June 1945 went unexpectedly airborne for over an hour after taking off with a limited fuel load.
The interaction between the propellers created an early stall condition forward of the wing when combined with the normal upward flow, causing the lift over the center wing to predictably suffer.  According to NAA aerodynamicist Ed Horkey:



  "What was happening was that we had propellers rotating in different directions on the left and right engines.  For some reason, which I can't remember, we started with the blades moving upward in front of the center section.  What this does, particularly at high angles of attack, is to create upward flow approaching the leading edge of the center section of the wing.  You also have normal upflow ahead of the wing.  The two upflows would add together to create an early stall.  The center section represented a large portion of the wing's area.  What was happening was that we were stalling out early and just not getting enough lift. ... We changed the rotations to go down the middle and we had the problem solved."