Can this part of the slayer exciter circuit be disconnected?
That circuit relies on the wire in question to create oscillations to drive the transformer.

Without that wire the transistor would never switch off.  There would always be DC current flowing through the primary side of the transformer.  Since the current wouldn't change, there would be no coupling to the secondary (high-voltage) side.

The LED's primary purpose is to make sure that the transistor base isn't pulled too far under 0 volts.  A standard diode would also work, but this way you get an visible indicator for free :)

There is a great answer here that goes into more detail about why the circuit works.