The solar system is non-integrable and has chaos.
The sun-earth-moon three-body system might be chaotic.
So, how far into the future can we predict solar eclipses and/or lunar eclipses?
How about 1 million years?
Answer
A number of studies have shown that the inner solar system is chaotic, with a Lyapunov time scale of about 5 million years. This 5 million year time scale means that while one can somewhat reasonably create a planetary ephemeris (a time-based catalog of where the planets were / will be) that spans from 10 million years into the past to 10 million years into the future, going beyond that by much is essentially impossible. At a hundred million years, the position of a planet on its orbit becomes complete garbage, meaning that the uncertainties in the planetary positions exceed the orbital radii.
What one can do is forgo the idea of predicting position and instead ask only about parameters that determine the size, shape, and inclination of planetary orbits. This lets one look to secular chaos as opposed to dynamic chaos, which in turn lets attempt to answer the key question, Is the solar system stable?
The answer to this question is "not quite". The key culprit is Mercury, the most chaotic of all of the planets. One factor is its small size, which magnifies perturbations from other planets. Another factor is resonances with Jupiter and Venus. Both of these planets have multiple resonances with Mercury's eccentricity (Jupiter more so than Venus), and Venus also has multiple resonances with Mercury's inclination. These resonances spell doom for Mercury. Mercury is perched on the threshold of secular chaos, and is likely to be ejected from the solar system in a few billion years.
The issue of chaos becomes even more extreme when trying to predict eclipses, particularly solar eclipses. The Sun, Jupiter, and Venus have marked effects on the long-term behavior of the Moon's orbit. Even more importantly, however, the Moon is receding from the Earth due to tidal interactions, and this rate is not constant. The current recession rate is about twice the average rate over the last several hundred million years. Changes in the shape and interconnectivity of the oceans drastically changes the rate at which the Moon recedes from the Earth. The melting of the ice covering Antarctica and Greenland would also significantly change the recession rate, as would the Earth entering another glaciation. Even a small change destroys the ability to make long term predictions of the Moon's orbit.
NASA developed a pair of catalogs of solar eclipses, one covering a 5,000 year spanning from from about 4000 years ago to about 1000 years into the future, the other, a 10,000 year catalog of solar eclipses spanning from from about 6000 years ago to about 4000 years into the future. The accuracy of these catalog degrades drastically before 3000 years ago and after 1000 years into the figure. Beyond this inner limits, the path of the eclipse over the Earth's surface becomes markedly unreliable, as does the ability to determine whether the eclipse will be partial, total, annular, or hybrid. At the outer time limits of the longer catalog, whether an eclipse did / will occur begins to become a bit dubious.
Because of the Earth's much larger shadow, predictions of lunar eclipses are a bit more reliable, but not much. The problem is that of exponential error growth, which is a characteristic of dynamically chaotic systems. Predictions of lunar eclipses more than a few tens of thousands of years into the future is more or less nonsense. The millions of years asked in the question: No.
The technique of orbital averaging once again can be of aid in determining characteristics of the Moon's orbit (but not position on the orbit). This can be augmented by geological records. Various tidal rhythmites give clues as to the paleological orbit of the Moon. A few rock formations exhibit layering that recorded the number of days in a month and the number of months in a year at the time the rock formation was created.
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