Besides the obvious cases where I'm behind a "one-way" mirror or have goggles/glasses on: is there one where I can see someone's eyes, and they can't see mine?
Fermat's principle says that the direction of travel for any light ray can be reversed. Therefore there is always a line of sight between a pair of eyes in both ways.
If one person is in the dark, then only one person can see the eyes of the other. So there needs to be enough light reflected from both person's eyes for this to work.
The answer of Martin Ueding is correct if there is no intermediate image in the light path. For example, if you use a camera obscura, in general there will in be no way for the observed person to create an image of your eye.
So the answer is NO for direct light paths and YES if you allow intermediate images.
Profiting from the fact that $c$, the speed of light, is finite, one could build a "monodirectional telescope" by using two shutters separated by some distance $L$.
The shutters stay closed for a time of $2L/3c$ and open for a time of $L/3c$, the opening time of the second being delayed by $L/c$. The photons entering the telescope from the first shutter will find the second open, while from the opposite direction it will be closed.
The fact that $c$ is fairly large does not make things particularly easy or practical, but it would work. Note that a similar apparatus can be (and has been) used to measure the speed of light.
Martin is correct, but neglects the case of distance. If you use a telescope or binoculars, you may have line of sight with the other person, but the other person may be too far away to "see the whites of your eyes". This is a function of how well you can resolve distant objects.
I'm not going to use the "observation by electronic camera" answer since in that case you are not actually seeing their eyes but rather a representation of their eyes on your monitor.
High power lasers use a contraption, called Faraday cell, or optical isolator, that allows propagation of light in one direction, but not the other.
The device consists of a Faraday rotator and two polarisation filters. The rotator uses Faraday effect, which rotates polarisation of light under magnetic field in suitable medium by angle depending on strength of the magnetic field in direction of the propagation of light, so returning light is rotated in opposite direction. Around the rotator are two polarisation filters rotated by 45° and the rotator is adjusted to turn the light by 45°. In one direction that allows the light to pass, but in the opposite direction the light arrives at the second filter 90° out of plane and is absorbed.
The wikipedia article also describes polarisation-independent variant that uses birefringent wedges instead of filters. In one direction, the light is properly recombined, in the other it is diverged and blocked by a collimator.
In the lasers, it is used to prevent reflections going back to earlier stages and causing additional pulses or even damaging those stages—high power laser is composed of oscillator that creates the initial pulse (say, 0.5 ns long) and several progressively larger amplifiers and the early stages are not designed for the powers at the end of the optical path.
The Faraday isolators are also used in optical communications.
The power lasers also use other elements, Pockels cells and Kerr cells. Both are blocks of suitable material that rotates polarisation only when electric field is applied. The cells are again guarded by polarisation filters so the light can pass only if appropriate electric field is applied. The rotation is reciprocal here, so light can pass both ways, but it is used to quickly open and close the optical path.
Usually a Pockels cell is used to split the laser cavity until the medium is charged and then connect it so the pulse can start building up and then a Kerr cell, which is faster, but needs (much) higher voltage, is used to let the beam out for the desired 0.5–1 ns.
This allows to build the device suggested in DarioP's answer measuring about 1 m.