I am not an expert on orbital mechanics, but I can suggest a possible situation in which a star system has some three dimensional orbits.
I can point out that from one point of view all the planets, asteroids, comets, etc. in our solar system have 3 D orbits. They orbit around the Sun, each orbit being apparently 2D, while the Sun orbits around the center of the Galaxy in a 200,000,000 year or so long orbit. And the plane where the Sun orbits and the plane of the planetary orbits are very much non co planer. So instead of moving in closed elliptical orbits, the planets, etc. in our solar system actually movie in orbits that are open helices from the point of view of the galaxy.
And there is another factor that actually makes the orbits of all bodies in the Solar System slightly 3 D. The planets, asteroids, comets, etc., don't all orbit in the same plane. They orbit in planes that are very closely aligned but still slightly tilted in various directions compared to other orbital planes.
And because the orbit of another planet is tilted slightly with respect to Earth's orbit, half the time that other planet will seems to be In a direction that could be arbitrarily called "above" Earth's orbital plane, and half the time that other planet will seem to be in a direction that could be arbitrarily called "below" Earth's orbital plane.
And because Earth and that other planet are attracting each other with gravity all the time, they are constantly "pulling" on the plane of each other's orbit and tilting the orbits a little more toward each other. But since their relative positions are constantly changing as they orbit the sun separately the relative directions of that pull are constantly changing.
And because the mass of Jupiter is much greater than the pass of all other planets combined, the ever changing direction and distance to Jupiter is the main factor that determines how much force in pulling on Earth and in which direction, but the changing directions and distances to the other planets modify the strength and direction of that force.
And so the tilt of Earth's orbital plane compared to the equatorial plane of the Sun is constantly changing very slightly in one direction or another.
Now image a solar system with two stars, and planets orbiting each star, instead of orbiting both stars. When planets orbit one star in a binary system they are said to have "S-type" orbits instead of "P-Type" or "circumbinary" orbits. In which planes will the planets orbit? The planets of each star in "S-type" orbits will orbit in planes tilted only slightly to the equatorial plans of their stars, since the vast mass of the equatorial bulges of the stars will gradually pull the planetary orbital planes closer and closer to the equatorial planes of the stars.
So can two stars in a binary system have different equatorial planes? If the two stars formed in the same collapsing protostar nebula then they are likely to have very similar equatorial planes which should also be very similar to the orbital plane in which they revolve around each other. But possibly various factors can make the three planes varying by much more than a few degrees.
Stars tend to form in vast collapsing nebulae that contain many smaller collapsing protostar nebulae that form the individual stars. And after all the stars are formed they make up an open star cluster, which gradually dissipates over hundreds of millions of years as the gravity of other stars pull the cluster apart. And while the new stars are orbiting each other in a relatively dense star cluster for tens and hundreds of millions of years there are any opportunities for gravitational interactions to make two stars capture each other and form a binary star.
In any case it seems that it might be possible for the orbital plane that two stars in a binary orbit around each other in to be very highly tilted compared to the orbital plane of any planets in S-type orbits around either of the stars. In some cases the orbit of the other star could be tilted approximately 90 degrees compared to the orbit of a planet around its star.
And thus the gravity of the other star, many times more massive than Jupiter, should pull the orbital plane of a planet "up" on one side and thus "down" on the other, since it has to stay centered on its own sun. But since the years of the planet as it orbits its star close in will be many times shorter than the orbital period of the other star several times farther out, when the planet moves to the other side fr its orbit the other star's pull will now pull the other side of the orbital plane "up".
Thus I picture the orbital plane of the planet wobbling from side to side, going in one direction for half a planetary year and then going in the other direction for the other half of the planetary year.
Any other planets that orbit the same star as this planet will also experience wobbling of their orbital planes, but to greater or lesser degrees, the lesser degrees closer to their star, and the greater degrees farther from their star.
And I think that it might be possible, under some conditions, for some or all of the planets that orbit Star A to have their orbital planes do a full circle around Star A at about 90 degrees to the equatorial plane of Star A. Thus each planet might have an orbit around star A lasting one planetary year in the planet's orbital plane, and also a much longer orbit of its orbital plane as that orbital plane rotates around Star A at an angle of about 90 degrees.
And to me that seems like it might be considered a three dimensional orbit.