solar system - What is the effect of the axial precession on the orientation of the planet's orbits?

When talking about the longitude of ascending node you must be very careful to define the reference plane that you are using. As you state, due to the Earth's axial precession, the First Point of Aries moves along the ecliptic over ~26,000 years. This is because the celestial equator is slowly precessing about the ecliptic.

Now, in the case of the orbital elements of other planets, the longitude of ascending node is again with respect to the First Point of Aries. As your intuition suggests, the orbits of the outer planets won't be influenced by the precession of the Earth's equinox. This means that the longitudes of ascending node will not be constant over long time periods. If we suppose that the physical orbits are fixed (i.e., the planets are not perturbing each other), the change in the longitudes of ascending node will be entirely due to the Earth's precession. When you take both these changes into account, you will find that they cancel each other out and the position of these orbits remains fixed with respect to the distant stars.

You can see in the orbital elements provided by NASA that they are given with respect to the J2000.0 epoch. In other words, the orbital elements are provided for a single point in time, and to get the orbital elements today you must take into account the Earth's precession.

http://ssd.jpl.nasa.gov/txt/p_elem_t1.txt