What is the effect of the axial precession on the orientation of the planet's orbits? I am working on an online application to visualize the positions of the planets and to calculate their ephemerides (you find it here, but it is not complete yet). I do not have much knowledge in astronomy and I am learning new stuff every day while developing the code. A few questions came to my mind that I hope to get answers for here.

The first thing I do not quite understand is the effect of the earth's axial precession. I understand the way it causes the vernal point to move along the ecliptic within the period of the precession of about 25770 years. But since the longitude of the ascending nodes of each planet's orbit depend on the vernal point and remain almost constant over time it seems like all the orbits change their position relative to the stars at the same rate as the vernal point does. But I cannot believe that the orientation of the orbits is related to the precession of the earth's axis.

You can see the effect in the application: When you move the topmost time slider for years you can see the zodiac rotate and the orbits staying in place because the vernal point is fixed with the scale. Shouldn't the orbits move together with the stars? But then the longitude of the ascending nodes wouldn't be constant anymore... I am confused. Thank you.
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