Is there a loss of torque when extensions are applied to the socket end of a torque wrench?

You know, occasionally, I've heard claims such as these as well.  But, when you need to use an extension to torque a bolt you have to use it.  Anyway, I've thought about it and the only loss that I could see would be the torsional flexing of the extension taking a portion of the torque you intended for the fastener.

Since the rotational deflection of a shaft (i.e. the extension) is directly proportional to the length of the shaft, then I would recommend using the shortest extension possible.  Also, since the rotational deflection of a shaft is inversely proportional to the shaft diameter raised to the fourth power (i.e. large stiffness change for a small increase in diameter), then I would recommend always using extensions with a diameter greater than the bolt diameter.

If you apply torque to one end of a solid shaft, that torque will be transmitted without loss to the other end.  This is because there is no counter-acting force or torque applied anywhere along the shaft (eg, no friction).

The longer the shaft, the more rotational deflection of the shaft will occur, but the torque wrench is impervious to that deflection because it does not use the deflection of the extension shaft to determine torque.  It measures the torque that it applies to its square drive.

U-joints can be a definite problem.  The two-pivot design, when operated at a non-zero angle, causes the rotational displacement, velocity, and torque to fluctuate cyclicly twice per turn.  In automobiles where this would be a problem such as on front-wheel drive vehicles, a constant-velocity joint is used instead of a U-joint.

So yes, a U-joint definitely can cause erroneous readings or tripping of the torque wrench.  Check out the third reference for measured data.

Long shafts should not be a problem.  If a shaft bends sideways at all, it mantains a constant velocity (and hence no torque irregularities) since it is a continuous joint, not a two-pivot joint.

Sloppy square-drive joints on multiple extension shafts might introduce error to the extent that they begin to act like a universal joint.

Note:  Even with U-joints, if you can maintain a straight alignment with the bolt head, the torque error theoretically is zero.