tldr: No, once an object is rotating, its axis will remain fixed in the absence of external forces.
A typical helicopter's rotor experiences drag which would causes it to slow down (transferring angular momentum to the surrounding air). In order to keep the craft aloft, the helicopter's motor needs to provide a constant torque to the rotor. This causes the body of the helicopter to rotate in the opposite direction at an increasing rate.
Since it's difficult to pilot a helicopter while vomiting, a tail rotor is used to provide a reverse torque on the body of the helicopter to keep it steady. The reverse torque is adjusted to match the torque of the motor (slight changes in the reverse torque allow us to yaw the helicopter).
Note - Not all helicopters use rotors.
Unlike a helicopter, a Spacecraft operates in a vacuum and therefore does not experience angular drag on its rotating parts thus no torque is required to keep it spinning.
The gyroscopic effect (a consequence of conservation of angular momentum) ensures our axis of rotation remains fixed. This is very useful for real-world spacecraft if we want to keep their solar panels and antennae pointing in a constant direction. It is also useful if we want to spin our entire vessel - as with a gravity ring.
Unfortunately, the real world often gets in the way of idealised physics. Here are some problems you may encounter:
Our ideal model assumes that there is no (net) external force on our
spacecraft. In deep space this is a good approximation, but things like
solar winds, atmospheric drag and tidal forces can produce a small net
torque. This can result in torque induced precession where your rotation axis vector traces a circle in space.
Unless your rotation axis is perfectly aligned with one of the
principal axes, you will experience torque-free precession. This is a constant periodic 'wobble' that can be seen when throwing a frisbee. This
becomes virtually guaranteed we consider that a manned spacecraft
will always have its internal mass distribution changing - people
moving around inside will change the moment of inertia and cause an
If you choose to spin on the wrong principal axis, you will encounter
the intermediate axis theorem and your spin will be unstable,
causing the rotation axis to flip periodically. This video provides an excellent demonstration of how to get spacesick.
If our craft has a rotating part and a non-rotating part, friction at the joint will conspire to spin-down the former and spin-up the latter.
All of these can be mitigated either with very careful design (see spin stabilised space probes), active management of the mass distribution or active attitude management using on-board thrusters.
Counter-rotating parts can be used to avoid the first three problems by having zero net overall angular momentum, but it still suffers badly from the fourth issue of friction.