Are the nearby airplanes cleared of the sky when launching Falcon Heavy?
SpaceX launches from Cape Canaveral Air Force Station (CCAFS), Vandenberg Air Force Base (VAFB), and Kennedy Space Center (KSC). All three have Restricted/Warning areas overhead and out to sea, from the surface to "unlimited" (i.e. space). It is illegal to fly through active Restricted airspace without permission. Warning areas are used over international waters and thus flying through them is not illegal, but they give fair warning you may not survive the experience, so pilots treat them the same.

These areas aren't active full time, but since activation of these areas is unpredictable (most have fixed schedules), airlines never plan flights to pass through them; it's simpler to fly around them every time, even though most of the time it isn't necessary.

GA flights, though, are quite common in those areas when they're inactive; ATC will allow flying over the launch areas, landing facilities, etc. as low as 500ft. When they're active, it's not uncommon for dozens of GA planes to be circling just outside the area to get the best view of the rockets, both going up and coming back down. However, GA flights don't show up on most flight tracking web sites by default, so you may not be seeing them.