Jeffbert, your linked article says essentially nothing about how the sudden lack of the Moon would affect Earth now—all the factors mentioned concern the development of the Earth up to now over geological and evolutionary time scales.
The epilogue of the pilot episode, "Breakaway," shows the Alphans gathered around the Main Mission monitor listening to a newscast from Earth. The anchor is reporting massive quakes and names three areas specifically (in the US along the San Andreas, Yugoslavia and southern France). He fades away into static as he reports on an emergency meeting of the International Lunar Commission to determine if anything can be done to rescue the surviving Alphans.
After the quakes, the sudden loss of the Moon would result in lower tides, as they'd be driven by the Sun, only. Some waterfronts might have to be rebuilt, but ocean-going traffic would continue. The greatest change might be a turbulent readjustment in the weather. The Sun dumps a great load of energy into the oceans, and that energy is then dissipated in various ways that affect land. I'd have to consult an expert on the subject, but I imagine the loss of the Moon would change something of that energy economy. Would the weather be more turbulent due to "hot spots," or would it become milder without the Moon's daily churning?
One other observation: None of the episodes ever mention where Alpha is located, although I've heard "writer's guide" info that it is located in Plato (far to the north on the edge of Mare Imbrium). This would put Earth fairly low in Alpha's sky, so the VFX are good on this point.
We also do not know exactly where the waste dumps are located, except that they are somewhere on lunar farside, which the opening titles erroneously call "the dark side of the Moon"—one would think the writers could have gotten a few facts straight before creating this fantasy. Anyway, if the dumps were only a little way over the pole, for example, the catastrophe might have given the Moon some significant spin, as the Moon is not neatly balanced, to say nothing of the vector crossing that of the Moon's orbit. If the dumps were farther over lunar farside, the Moon might even have plunged closer to Earth on its way out. Booyah.
The "gentlest" ride would come from the dumps being squarely on the equator in a location to complement the Moon's orbit, rather than cut across it, or even thrust the other way.
The one thing I will say in the show's defense is that most critics of the "Breakaway" scenario assume a single explosion. Given one big enough to push the Moon out of orbit, it would probably destroy the Moon altogether. However, Prof. Bergman reported the dumps as acting like a giant rocket motor as they burned out. Even this scenario is sketchy, as the thrust would probably have to run for days, at least, just to break orbit, to say nothing of the interstellar speeds later attained.