Do the front tires of a car act like gyroscopes, such that a car could steer on a frictionless surface?
No, a car cannot steer on a frictionless surface. This has little to do with gyroscopic action and more to do with conservation of momentum: to turn, even when conserving its speed, the car needs to accelerate at right angles to its motion, which changes the total momentum of the motion. This change in momentum requires a force which, in normal roads, is ultimately provided by the friction between the tyres and the road. In the absence of friction, the car tyres would skid sideways with respect to their rotation (i.e. along the axis) without being able to influence the car's inertia.
It's important to note that, because of the gyroscopic effect, the car can indeed change the direction it's facing pretty much arbitrarily. The easiest way to accomplish this is to have a big flywheel, with a horizontal axis, inside the car, with a mass that's at least comparable to the car's. If you then try to turn the flywheel's axis within the car, you will instead turn the car around the flywheel, because of conservation of the large amount of angular momentum in the flywheel. (This will also cause a torque on the car about a horizontal axis, but this can be cancelled by the normal force from the surface.) However, even if you manage to turn the car 90° from its direction of motion, it will continue to move in the same direction as before, with its wheels skidding perpendicularly across the ice.
Also, as other answers have mentioned, if the car can interact with the air in any meaningful fashion - either by its air intake and exhaust, or by using its bulge as a sail, or by propping up an actual sail - then it will indeed be susceptible to external forces and it will be able to change its direction of motion. Similarly, the car would be able to steer if it could chuck rocks, bump off of other cars, or use rocket thrusters. I don't think this directly answers the core of the question, though.
If the wheels had spun fast enough for a gyroscopic effect to become noticeable, the only result on a frictionless surface (which would be the same without a surface at all) is that when you turn the wheels, the rest of the car would rotate instead of just the front wheels :)
You need some reaction force to alter the trajectory, like a sail or surface friction or thruster.
Yes you can
It is actually possible with a real car, but you would have to be very patient to steer a little bit.
Suppose you have built a car with power on the big front wheels to induce a gyroscopic effect. If you rotate the wheels, the direction in which the center of mass is going will not change directly, but the angle in which the rest of the body points will change.
Now we use another property of cars: Often they do their air intake in the front, and have the exhaust in the back. This results in a net force that is roughly pointed towards the nose of the car. So, as you can turn the car a bit by turning the wheels, you change the direction of this force, and will eventually be able to move the car a bit left or right.
Since there is no friction, then it will not affect any other forces that may act on the car.
The direction of wind blowing on the car may change its trajectory, as any driver will attest when driving in high winds. Turning the car wheels may have a slight affect on the resultant direction of the force.
If the car has curved roof, then it may acts as wing. Getting the car onto two wheels (ie. by hitting a rock on the road) allows you to change the direction of lift, and you could steer the car in a similar manner to how you steer a plane.
If you opened the car window, and threw something out (or shot a gun), then Newton's third law would move the car in the opposite direction.
The exhaust pipe acts as a jet rocket. If you attach a hose, you could point it in different directions and steer the car this way.
If you stuck a sail on the roof, you could control it by wire with the wheels, and steer like you would a boat.
Friction is the only force that would cause the car to move along a different path. On a frictionless surface, the gyroscopic effect could change the orientation of the car a bit, but not the trajectory of the car. In other words, the front car would no longer point along the direction of travel, but would "skid". (That is, if you could call frictionless sideways movement "skidding.")
On a completely frictionless floor, with the absence of other external forces, the centre of mass of the car will continue in the same trajectory for ever. Hence no steering is possible.
However, irrespective of whether the front wheels are rotating or not, turning of the front wheels will produce a counter torque changing the orientation of the car, albeit by a very small amount. The changes in the orientation of the car and the tyres will be in inverse ratio of MOI(moment of inertia) of the car and the tyres around the wheel steering axis.
Regarding the gyroscopic effect, since the wheels are rotating around an axis parallel to the ground and the turning of the wheels is done around an axis perpendicular to the ground the gyroscopic effect will be experienced around an axis perpendicular to both of these. This axis would also be parallel to the ground. Hence assuming the gyroscopic effect is small owing to the small angular momentum of the wheels, the only effect would be redistribution of the weight of the car over its different wheels. If the front wheels were rotating sufficiently fast the gyroscopic effect would produce enough torque to topple the car over to one side.