Why does Uranus not experience gyroscopic effect? As I understand, gyroscopic effect is when a body has multiple distinct axes of rotation. This video shows this effect when a body already rotating in one axis is made to rotate another axis perpendicular to the existing one.
If the rotational axis of Uranus is perpendicular to its heliocentric revolution, why has this not had any apparent effect on the planet's orbit?
 A: The orbital motion of Uranus around the sun is not a rotation, so there's no issue.
The video shows the neat effects when something tries to change the angle of the axis.  It's not shown, but if the person holding it were to just walk in a circle while holding it in one direction, it wouldn't be any more difficult than holding it while standing still.
Likewise, for the most part the orbit of Uranus around the sun has no effect on the rotation because it's acting uniformly on the entire body, not creating a torque or turning motion on it.
Take a table gyroscope and compare trying to change the angle of the axis, and moving the base in a circle on the table. 
A: I think BowlOfRed's answer is a sound first way to think about the question and it's certainly true in the short term for a small enough gyroscope. Another slightly different way of looking at the problem is to understand that that any short term precession of a small enough gyroscope would tell against the Equivalence Principle, as any precession of such an instrument would otherwise be a way of detecting freefall in the Sun's gravitational field.
However, in the long term, a much fuller analysis would be needed. Certainly even small gyroscopes in orbit around a gravitating body do precess over the long term (through the Lense-Thirring effect, aka "Frame Dragging") as confirmed by Gravity Probe B tests. Even if we restrict ourselves to Newtonian gravitation, Uranus is wide enough to feel a very small tidal effect from the Sun's gravitational field, which tidal effect might transfer angular momentum between the Sun and Uranus. So, without further analysis, one cannot say what the long term effects on the rotation would be.
