This was an interesting subject..
Initially my hunch was with ChristiaanJ, but as of now I am not sure if I completely agree.
Since there is much discussion about definitions in this thread I will state my own. I am certainly no aeronautical engineer, but I see these definitions as a prerequisite for the discussion.
I will use thrust (T), drag (D), lift (L) and weight (W). We have a vertical and horizontal axis on our aircraft through the center of gravity. Weight is always directed towards the center of the earth. Thrust and drag are for simplification aligned with the flight path, and lift is directed perpendicular to the flight path. I will also call the angle between the flight path and the horizontal through the center of gravity the climb angle (a).
To then sum up the positive vertical vector during climb we have:
T*sin (a) - D*sin (a) + L*cos (a) - W
This would give a fixed lift for any given speed regardless of pitch-angle and thus the induced drag will be the same as for that specific speed in horizontal flight.
The negative component of drag ( D*sin (a) ) and reduced vertical component of lift ( L*cos (a) ) must from what I see be countered by the increased positive component of thrust ( T*sin (a) ).
So.. neither increased nor reduced induced drag during climb.
As stated earlier I am only a layman on this subject, so any corrections making me wiser are very welcome.