Hey, I recently bought a Tascam 644 Midistudio, and I have the exact same problem. Pressing the play button engages the machine for only a couple seconds before it stops automatically.
Now I have discussed this a friend, who is very much into 4 tracks tape players, Tascam and such. He told me he had the same issue with two machines (with a smaller model tho, a 424 iirc). He tried changing the belt, motors etc, to no avail. Then he found out the problem came from the power unit. He bought a new one, and both machine then worked fine.
If you can find a power supply, maybe that is something you can try. I didn't bother because the 644 has a weird kind of power supply and power input. So I would need to buy it new on ebay or something, too expensive.