When there's no tape to hold the motors back and the switch in the tensioner in engaged, the reel motors should both turn in play. The left one (supply side) should turn clockwise while the right one (take up) counterclockwise (creating back tension).
The capstan should turn when the tensioner is lifted. If it doesn't do this, it could be a problem with the motor control board or the power supply.
I'd start troubleshooting by downloading that manual and being sure that all the setting are correct and check the internal fuses. Be sure that all the editing controls are off. They may actually disable the motors.
Do all the following with the machine turned off. If you don't feel comfortable with electro-mechanical devices, you may want to have someone else do the following.
The easy stuff would be to check to be sure all the multipin connectors are seated correctly. Carefully pull all of them you can easily reach (sometimes you'll need to loosen them with a flat bladed screwdriver,.. careful though, sometimes those circuit boards can become brittle with age and cracking one just increases your problems). Use a light and examine the pins that the connector goes on. If they aren't nice and clean, use a little contact cleaner on a q-tip before you put the connector back on. Take your time and try the machine with each one.
If you can tell which is the power supply board, start there.
I believe that your Otari has edge connectors on some of the circuit boards. You might try wiggling the boards around a little to seat them a little better. When a machine sits around unused for a while, those edge connectors can get oxidized so shifting them around in their connectors can make a clean connection.
While you're in there, try turning the capstan motor by hand. It should only be impeded by the weight of the motor/flywheel.
Well, give those things a try and let us know how it goes.
Also, any descriptions of noises that the machine makes when you try and play a tape might be helpful (does it sound like the motors are trying to turn etc).