If I'm looking at the correct circuits - the resistor is R1310, a 1Kohm, connected between the base of V1310 (a PNP BC808) and pin 6 of connector P1530/N1530. When pin 6 of the connector is shorted to ground (by a microswitch on the tape deck) the transistor will conduct causing the mute circuits to activiate. The microswitch should be open circuit in normal tape operation. There is no resistor shown to pull the base high or keep the transistor "off", so the base is indeed floating during normal operation.
Lifting R1310, or disconnecting the wire into pin 6 of P1350, will have the same effect - and will stop the mute circuit from working. However, if that works it's not a leaky transistor - it's a problem with the wiring to the connector or the microswitch. However, it might be worth wiring a high value resistor (10K or more) between pins 3 and 6 on the connector. Pin 3 should be a stable 5V supply, and 10K should be more than enough to ensure the transistor (V1310) stays off when it should be off.
PS - I'm dying with man-flu at the moment, so might not be able to add any more unless I make a miraculous recovery.