Ah, sorry about that. I didn't catch the model number for your machine and I mistakenly assumed it was an RT-707. There is no reversing relay in an RT-701 of course.
One thing you should do is to thoroughly clean the contacts in the record switch (not the two record safety switches up top, but rather the main record button next to the play button). This button has contacts which enable the pitch control in play mode and override it in record mode. If the contacts in this switch get dirty and loose continuity the speed can be affected. Does the speed run away in the the record mode too or only in the play mode?
If it runs away in both modes then check the frequency generator connections at the servo board (small wires leading into the motor) since the motor may be loosing it's reference signal.