This seems to be very similar to the myth of NASA spending millions of dollars to develop a space pen. One thing is certain: This did not happen (quoting from tutorvista.com):
Voyager 2, in 1986, as it flew past the planet Uranus, was set into unwanted rotation by this flywheel effect, every time its tape recorder was turned on at high speed. The ground staff at the Jet Propulsion Laboratory had to program the on-board computer to turn on counteracting thruster jets, every time the tape recorder was turned on or off.
Even worse, (from education.com),
What Caused Voyager to Point in the Wrong Direction?
While the second statement in that tutorvista.com link is true (see Hobbes' answer), the first is not. The title of the page at education.com is even worse; it's completely bogus.
At the time of the Uranus encounter, it took 2.5 hours for information sent from Voyager 2 to reach the Earth. It would have taken an additional 2.5 hours for any corrections to have been sent back to Voyager 2 -- and that's assuming the engineers could have instantaneously recognized the problem developed a plan to compensation for that undesired rotation. The Uranus encounter was of a very short duration, much like the recent fly-by of Pluto by the New Horizons spacecraft. The bulk of the encounter would have been already been over with that (minimum) five hours time between problem recognition and problem resolution.
In fact, it would have taken much longer than five hours. Propellant was the one of the most precious resource on the Voyager spacecraft. A plan that involved expending propellant would have had to have gone to the highest level for approval. The time frame needed just for this approval would have involved weeks, perhaps months. It would also have taken a good amount of time to develop the compensation algorithm. This was not a seat-of-the-pants decision.
There's another problem with the entire premise. Quoting Donald Rumsfeld, "There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don't know we don't know." Spacecraft attitude control systems need to deal with the known knowns, the known unknowns, and the unknown unknowns.
Had they been uncompensated, the attitude rate errors that would have resulted from starting and stopping of the tape drive would have fallen into the category of "unknown unknowns" ("process noise" in terms of Kalman filters). The attitude control system would eventually have addressed these errors, but only after the fact. The result would have been fuzzy images rather than pointing in the wrong direction. Addressing these knowable consequences of starting and stopping the tape drive as an integral part of the attitude control system moved this from the "unknown unknown" category to the "known known" category.
Perhaps this was foreseen during the design stage, perhaps not. It's hard to tell; I searched and searched. At the very minimum, someone at JPL foresaw this months in advance of the Uranian system encounter.