I have seen the same behavior as the original poster even with a STAR loaded.  In fact last weekend on a flight South to North across Australia.  Lateral navigation wise she flew the STAR just fine.  There was no T/D and no descent profile however.  FMC showed FL360 all the way to the IAF for the ILS.

Also the  arrival time at each waypoint on the flight plan was shown as the current time and the fuel remaining did not decrease on the FMC.  I took off with 26000KG fuel and for each waypoint including the destination airport the FMC showed 21.1 (i.e., 21100KG) remaining.  The fuel readout on the overhead panel was fine.

Good thing I can do the calculations manually.

I did use a derated T/O.  Does that mess up the FMC?