Now you know that this can not be done for SO MANY reasons, but as a simple maths exercise it is straightforward.

We know the rotation period of the moon (~2360600 seconds).  We know the radius of the moon (~1737400 m).  Therefore the speed of rotation at the moons equator is 2*pi*radius/rotation period (~4.62442 m/s).

Light speed is 299792458m/s.  If you now divide the speed of light by the speed of rotation you get a factor of 64828119.  So your tower would need to be 64828119 times the radius of the moon, minus the existing radius of the moon of course.  I calculate your tower at 112632372091547m.  More the 750 times the distance to the Sun.

Speed relative to what?

EDIT: Sorry. That's not a very useful answer. What I mean is that you can't really start to work out the answer unless you know what to measure against. If we agree that the top of the tower can't move at light speed, because it breaks the laws of physics, let's agree to get it moving really, really fast.

So build a tower and measure its velocity with respect to a distant galaxy, and surprise surprise, it's already moving at a large fraction of the speed of light. Measure its velocity with respect to the surface of moon and it isn't moving at all.

Talking of fractions, I'd estimate that about one twentieth of the questions on this site contain a proposal for light-speed travel (or faster). Surely the FAQ section is long overdue?