Theoretically speaking, what orbit should the earth's moon must take so that there are never any eclipses - solar or lunar? Is it mathematically possible to construct such an orbit?
Answer: yes, a no-eclipse orbit is possible
The plane of the Earth-Sun orbit (the ecliptic) and the plane of the Earth-Moon orbit must intersect each other because they both contain at least one point in common: the CM of Earth. Two planes always intersect in a line, so there will always be two POTENTIAL positions for the three bodies to be aligned, as illustrated below.
However, that doesn’t mean the bodies MUST align: if a “month” was a proper fraction of a year, the Moon could always be above (or below) the ecliptic at times of potential eclipse.
For instance, consider an idealized Earth/Moon system with the Moon in a circular orbit of period 1/12 of a sidereal year and the Moon’s orbital inclination is 5* with respect to the ecliptic.
Lunar eclipses occur when the moon is in positions A or D. Solar eclipses occur on positions B or C. The line A-D is the intersection of the ecliptic with the Moon’s orbital plane. Eclipses can only occur when all 3 bodies are on this line.
If the moon’s orbital period is a proper fraction of the Earth’s orbital period, the moon’s position in its orbit will be synchronized with the Earth’s orbit. For instance, if the moon's orbital period is 1/12 of a sidereal year and it starts in position E, a half year later it will have completed 6 orbits of the earth and be in position F. The moon has “missed” the only two opportunities in the year to have an eclipse. And since its orbit around Earth is synchronized with the Earth’s orbit around the sun, it will never get the chance in future years.
Answer: Yes, there are multiple no-eclipse orbits (other than the phase-shifted period solution by @woody) possible if we leverage various perturbation models of the two-body system.
The J2 effect. If you take into account the gravitational effects of Earth's equatorial bulge on the orbits of satellites, this causes a precession of the any satellite's orbital plane around Earth's rotation axis. Satellites in polar orbits use this effect to rotate their orbital plane with a rate of one full rotation per year to view the Earth under the same solar angles permanently. We can use the same effect to keep the Moon's orbital plane from intersecting the Earth-Sun line indefinitely by placing the Moon in a polar low Earth orbit, at a range of altitudes with each a specific inclination which you can calculate with the approximating equation given in the Wikipedia article on Sun-sync orbits. However, as the Moon has a radius of 1736 km and we'd rather not have it scrape across our crust or burn up in the atmosphere, we should prefer orbital altitudes above 2000 km. As solutions to the combination of altitude and Sun-synchronous inclination exist only below 5981 km altitude, the longest orbital period that this kind of non-eclipsing Moon will have is 3.8 hours. Get ready for an Interstellar-like tsunami world!1
Lagrange Points. If we look beyond a strict Earth orbit towards the Sun-Earth two-body system, there are five locations where the gravitational attraction of both bodies and the centrifugal acceleration cancel out in the frame of reference rotating with the Earth around the Sun. In these so-called Lagrange points you could place the Moon to make it follow the Earth on its orbit around the Sun, without eclipsing it. Your best long-term choices are the stable L4 and L5 points, where the moon can sit still or librate in small loops around one of them. Unfortunately, L1, L2 and L3 are unstable, so if you tried placing the Moon in a halo orbit (to avoid the eclipse cases) around them, it would soon end up drifting off into the unknown. A disadvantage of L4 and L5 is that the Moon will appear rather small and create insignificant tides as it would be 389 times further away from Earth than it currently is.
1 Your Moon won't last long in this orbit though, as tidal forces or centrifugal forces if you choose to let it rotate with a 3.8 hour period will rip it apart.
Edit: props to @AJN and @Greg Miller, just saw that they had already proposed these ideas in the comments
Edit2: added Roche Limit and clarified Lagrange case
Edit3&4: corrected Lagrange point formulation