I can't seem to find any info online re: the eccentricity of the orbital plane of the Earth.

I know of the 26,000 year wobble, and the orbit is becoming more eliptical, I believe. But does the plane of orbit ever go up or down?

In other words, will the ecliptic ever change its path in the night sky, or will only the background stars change slowly?

The quick answer is that the plane of the Earth's orbit does change, just very slowly.

In the Solar System there is a plane, called the "invariable plane" or the "Laplace plane", that is perpendicular to the total angular momentum of the Solar System. This plane (the invariable plane) does not move. Jupiter makes the largest contribution to the total, so this plane lies close to the orbital plane of Jupiter. The Earth's orbit is inclined to the invariable plane by about 1.6 degrees. I forget how fast the node of the Earth's orbit on the invariable plane precesses, I think it is clockwise (retrograde) once every 70,000 years. (Someone please correct me on this if necessary.)

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Useful search terms are "planetary precession" (the old term for this effect) or "precession of the ecliptic" (the newer term approved by the IAU).
There's a diagram here showing the location of the pole of the invariant plane, and the ecliptic pole at two different dates. Motion of the ecliptic pole is anticlockwise as we look at the sky, so clockwise when observed from outside the solar system, as Celestial Mechanic says.

Grant Hutchison

The invariable plane is almost invariable, much as the solar contant changes from about 1375 watts/ square meter during the year, and is likely increasing a few microwatts per century. The plane of Earth's orbit is three dimentional in parts per billion, both short term and over billions of years. That is to say the plane is slightly wrinkled. Precice measurement is difficult because the photosphere of the sun is thousands of kilometers deep and more or less transparent to both visable photons and radar photons. Neil

The invariable plane is invariable, as the name suggests.
It's the plane normal to the angular momentum vector of the solar system, which will change only if we acquire a new body orbiting the sun.
The ecliptic plane is defined according to the orbit of the Earth-Moon barycentre, and it does precess slowly with respect to the invariant plane, as described above.

Grant Hutchison

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Well thanks, everyone else, for your responses. Figure 11 of THIS RESTLESS GLOBE answered my question of where was the ecliptic 4000 years ago (not far from where it is now). Of course I'd like to see where it'd be in another 4000 years. Can anyone think of a software that would show me the ecliptic and celestial sphere as they have changed/will change in the past/future?

I guess the inclination of the ecliptic in relation to the invariable plane alternates between 2.6 and 0.8 degrees, with a period of 100,000 years. But I guess nodes of the ecliptic (and the invariable plane?) precess or recess too. That's why it'd be good to see it visually. I'm having fun with the freeware, CELESTIA, but I don't think I could plot the movements of the ecliptic and celestial sphere with that. What about ASTROLABE? That has confused me in the past.

Here are some graphs I made with Gravity Simulator and Excel of the dynamic orbital elements of the Earth/Moon barycenter over a period of about 90,000 years.

These are with respect to the ecliptic plane, not the Laplace plane, but still it can show how Earth's nodes precess with respect to a fixed plane, as the ecliptic is defined as Earth's inclination in January 2000, and Earth's instantaneous inclination is free to drift.

The final plot is Earth's inclination over the period of 1987 to 2022, just to show that it reaches its minimum (1/10000 of a degree =~0 degrees) in the year 2000.

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