Other answers described how this would destroy both Earth and Moon in short order. Assuming the mad scientist wants to avoid that (it would destroy the tractor beam, those aren't cheap!) there is a way to limit damage.
The tractor beam is able to change the orbital and rotational speeds of the moon. This is necessary to have the Moon immobile compared to the Earth surface. Any of those changes has to be acted on the tractor beam itself, as per Newton's third law. Assuming the tractor beam to be on Earth, it is probably acting on Earth at the same time to avoid being shot to outer space like a cosmic ray.
With careful changes of Earth orbital speed, possibly pushing against passing asteroids or distant planets to gain or loose momentum as needed, you may end up with a special configuration: a contact binary planet aka. Rocheworld.
In this case, the Moon is orbiting around at normal orbital speed for its altitude (1737 km, Moon radius), which is a tiny bit more than 7 km/s, completing one orbit in about 1 hour 35 minutes.
To avoid friction between the Earth and Moon surfaces now in contact, Earth is orbiting at the same angular velocity. That is, one Earth day is now also 1 hour 35 min long. Which itself would have some interesting effects on climate and geology (and by "interesting" I mean "apocalyptic")
That way, the Moon doesn't fall on Earth as it is in orbit, and is kept in one piece by its own gravity. Similarly, it isn't gouging Earth at the contact point because Earth and Moon are moving in concert.
This will still cause massive earthquakes as the pull of gravity suddenly changed intensity pretty dramatically at places. Both Earth and Moon will also slightly reshape, meaning apocalypse-level earthquakes and landslides pretty much everywhere. Atmosphere will also rush to cover the Moon, as atmospheric pressure is everywhere more or less the same on a Rocheworld, and there is suddenly a 1 to 0 atmosphere difference to equalize. Longer term, atmosphere starts evaporating on the Moon side due to the weak gravity,
I am not sure a Rocheworld is stable with such a big difference in mass between the two parts, so the Moon may end up partially break apart to form a ring and partially fall down to Earth, which would reshape back as a sphere. Or it may stay as it is, with enough ground moving to the contact point to make a big contact circle - as matter settles until everywhere points down, that is with gravity being perpendicular to the surface, the contact circle would point sideways compared to either center.
It would still kill about every living thing on Earth and flatten most of its features, but at least the mad scientist would have more time to pack the tractor beam on a spaceship.