Let's say that it's impossible for a hypothetical organism's proteins to denature - unfolding and loosing their structure - due to high heat, and that, as such, temperatures equal to those required for those proteins to denature no longer present a health hazard to said organism.
Additionally, let's say that this organism is an animal - one with an intestinal tract - and that its intestinal permeability scales with the body's thermoregulatory mechanisms; that is, when the temperature increases, the body decreases its intestinal permeability, preventing endotoxemia, and, therefore, also preventing sepsis.
Note that these things are handwaves. I'm going to soften them up a bit by claiming that this hypothetical organism also produces many more heat shock proteins - proteins designed to mitigate stresses on cells, including high temperatures - than normal, as well as producing high levels of cyclic 2,3-diphosphoglycerate and having lots of topoisomerase V (see below), but it's still a handwave.
The question: what's the next problem? After septic shock and protein denaturing are overcome, what's the next overheating-related thing to cause physical harm to/kill this organism?
Assume that, aside from those alterations, this is a multicellular, mammalian vertebrate that runs on Earth biochemistry - think "human".
I was inspired by this answer to a previous question of mine; in it, theorist points out that the bacteria M. kandleri has high concentrations of cyclic 2,3-diphosphoglycerate; moreover, its Wikipedia page points out that it is the only species to have topoisomerase V, which apparently enables it to survive at temperatures of up to 110 degrees Celsius/230 degrees Farenheit/383.15 Kelvin.