The reason I’m going with “not a tension issue” is the takeup motor jumps. If it was a tension issue it wouldn’t move at all. If it won’t go when there is no load (the tape is slack), it certainly wouldn’t move at the beginning when it is overcoming static forces, tension arm load, etc.
Not necessarily. The _PLAY logic line does a number of things, one of which is to 'flash' the reel motors, sending a burst of 24v to start them up, before settling to 12v - and I
think that bypasses the tension subsystem, so it's possible you'll get an initial kick even if the tension circuit has gone dead. This is just conjecture, of course - unlike the Studer or TASCAM 32 manuals there isn't really a "Theory of operation" section.
As for the rest of the _PLAY logic, I think it gets inverted by U15, goes through diode D11, with diode D12 possibly bringing in an input from _FF and _REW (inverted by U16). It then gets inverted again by U28, is attached to a 5v pullup resistor (R69). After that, the signal is split two ways, for the left and right reel.
One of them goes back into U28 to be inverted yet again, then through the R101 tension adjustment (4.7K by the way), through R41 (220 Ohm) and on to the reel motor drive op-amps and so forth.
The other signal does essentially the same, being inverted via U28 but then going though R102 and R42 respectively.
U28 is a 7-way inverter. Pins 1+16 are unique to the _PLAY logic, the other 6 inverters are used for each of the tension adjustments. Aside from the actual tension circuit itself (R101/R102, R41/42) the only unique component to the play/tension logic circuit I can see is U15, a DTA124ES, which I -believe- is being used as an inverter going by the schematic symbols and apparent behaviour, but frankly analysing this is hitting the limits of my knowledge. I can do digital logic, but my basic electronics are not really good enough.
I've been going by a print copy of the schematics I got off ebay, but since people are asking, I had a look and found this:
TEAC TASCAM TSR-8 Service Manual download, schematics, eeprom, repair info for electronics experts
...which should sort you out for the service documentation and schematics.
EDIT:
Ooh! Something else to try... what happens if you put empty reels on the machine and put it into play? You'll need to hold the right tension arm to convince it to run. I'd be interested to see if the left reel spins at all, i.e. whether we have back-tension during play.