At what point in the future before the heat death of the universe (which I believe will be in 10¹⁰⁰ years' time) will it become impossible for any sort of biological life anywhere in the universe to continue?

Will it be when all the stars have burnt out or what?

Cheerz GIAN 🙂

If proton decay is real, that would be the real endpoint.  Before that, there is baryonic matter which can be manipulated and from which energy can be extracted.  After decay, you just have a dispersed cloud of leptons and photons.  And thermodynamic equilibrium, where no work can be done.  I would search on "proton decay" to see what conjectures are for timelines of such decay.

Thermodynamic equilibrium is a probabilistic phenomenon. There is a VERY small probability of a VERY RARE arrangement of constituents particles. This probability is not 0 though, so if we wait THAT long, it will perhaps happen.