Dooood,
I have not had exactly the same thing, but very similar. My rig (borrowed) would appear to be recording, the levels were bouncing, etc. When i went to do the transfer, the rig would play for a second or two and stop no matter where in the tape. Most of the 4 (ouch) DATs to cover the 6 hour (ouch ouch) lost time coding as well, although the beginning of one was OK.
I know that SOMETHING was recorded on the tape, as i could get a second or two of audio in random places. Sent it to TASCAM and they knew exactly what was wrong and fixed it. The bummer is that i still have not found a way to rescue the other music. (Any help there???)
Anyhoo, the grease thing sounds plausible. Point of the story: get it serviced! And try really hard not to let them tell you the heads need to be cleaned/replaced (unless they do).
UJ