If there is a time's arrow (infinite aperiodic evolution), the descriptive complexity of the Universe will increase as a logarithmic function (in base 2) of time (here is the proof of my theorem). Does something that is increasing its descriptive complexity all the time deserve to be called dead? I do not think so.
Another possibility is that time is cyclic. In this case, the evolution of the Universe will be like the finite cellular automata studied by S. Wolfram, e.g., rule 30 and rule 90. After the heat death, there will be a big bang again.
Finally, there is the possibility that the Universe will halt and everything will be frozen forever.