I think that a solution might be possible. According to string theory our universe is likely to be only metastable since its cosmological constant is positive. It means that eventually we get false vacuum decay and the formation of a new universe. If the new universe has zero or negative cosmological constant, its asymptotic temperature will be zero which should (I think) allow avoiding heat death (that is, performing an infinite computation). Now, I think the half-life of spontaneous nucleation within a cosmological horizon is usually predicted to be much longer than the relaxation time. However, this leaves the possibility of heterogeneous (induced) nucleation. Now, I'm not aware of any research about artificially induced false vacuum decay, but I don't know of any physical barrier either. If we manage to induce such a decay and find some way to transmit ourselves into the new universe (which probably requires the new universe to be physically universal), avoiding death is a possibility.