Semi-classically there is no issue to look at late times of the QFT of the standard model in our nearly de Sitter looking space-time geometry. The contribution to the vacuum energy of matter and forces is a well defined quantity, and so is the cosmological constant; in this context it's logically conceivable that it just stays what it is, a constant (the vacuum expectation values of the scalars may have a time-dependent potential contributing to the vacuum energy, a bit like in inflation, in which case that could vary; cf Quintessence scenarios). This what the standard model of cosmology describes. Nothing special happens at late times.
There are however several reasons to suspect that de Sitter is not quantum mechanically stable in quantum gravity (or even metastable in the latest reboots of the "swampland conjectures" program) and that the universe could undergo some sort of contraction (due to quintessence for instance) at some point. So the universe wouldn't be evolving towards heat death but would heat up again. It's not possible at the moment to give a definite answer to this question as a proper theory of quantum gravity has not yet been found to exist in de Sitter space (nor rigorously disproved to exist).