The Earth rotates about it's own axis. Do the geomagnetic field lines rotate due to this rotation or not?
Yes, the Earth's magnetic field does rotate with the Earth. There is a simple way and a complicated way to explain this.
Firstly the simple way: the magnetic north pole and the North Pole are not at the same point. That means if the magnetic field did not rotate with the Earth the magnetic north pole would rotate once around the North pole every 24 hours. Since the position on the Earth's surface doesn't rotate every 24 hours that must mean the field is rotating along with the Earth.
But the Earth's magnetic field could still be rotating about the axis of it's dipole, even though that axis is fixed wrt the Earth. If this were the case then at a position fixed relative to the Earth's surface there would be a changing magnetic field and that would induce a current in any conductors like telephone wires, transmission lines and any bodies of salty water (i.e. seas!). This current would oppose the changes in the magnetic field and oppose the motion. The end result is that any rotation of the magnetic fied relative to the Earth's surface would be brought to a stop.
If you're interested in pursuing this further there's an introductory NASA article on it here.
Yes, the geomagnetic field does rotate with the earth. This is the reason why maps of the geomagnetic field overlaying geographic coordinates are reasonably accurate for a decade or two - and why a compass is still useful for navigation. (i.e. You do not need to know the time of day in order to correct for the magnetic declination cited on your map!)
The physical explanation for this is that the geomagnetic field originates due to electrical currents generated by convection in the earth's liquid outer core. The core rotates with the earth.
However, the geomagnetic field rotates at a somewhat different rate than the whole earth, and the magnetic and geographic poles do not coincide.
The geomagnetic secular variation becomes noticeable on maps older than decade. There is a prominent feature in the non-dipolar part of the secular variation that manifests as a westward drift at a rate of about 0.2 degrees per year.