After trying to answer this for a while, I've reached the conclusion that the information needed to answer it satisfactorily doesn't exist. I'll go ahead and summarize my findings here for you though, hopefully it's informative.
Different viruses infect different cells, and different people's immune systems are more or less successful at limiting infection, so the number of cells that get infected also varies a lot. A useful way to narrow your question might be "What is the maximum fraction of virally infected cells that can be survived?". All infections will vary between this maximum and 0%, with milder disease being associated with lower % cells infected.
I looked at Ebola as a candidate because it can infect a wide variety of cell types and a lot of research has been published on it.
From J Virol. 2004 Apr; 78(8): 4330–4341.:
Notably, 20 (91%) of 22 nonsurvivors had viral loads that reached ≥10^8 RNA copies/ml within the first 8 days after the onset of symptoms, thus suggesting that 10^8 RNA copies/ml can be considered an approximate threshold that predicts a fatal outcome with a positive predictive capability of >90%.
So, for about 3 L of blood serum in a person, that's 3 * 10^11 copies of the viral genome in the blood as a threshold for survival. I couldn't find useful information about viral load in organs and other tissue, but since blood has been claimed to have much higher concentration of virus than saliva or other fluids, we can assume this number is a significant portion of the total load. I'll assume about a third, so we get 10 ^ 12 genome copies. Some or even most of these RNA genomes might not be incorporated into viral particles but counting actual viral particles is very difficult, so no-one has published a correlation between Q-RT-PCR results and infectious particle counts as far as I could find.
Next, we need to estimate the number of progeny viruses that each infected cell produces. In Clin Diagn Lab Immunol. 2002 Jan; 9(1): 19–27. , researchers got about 2 * 10^4 PFU from wells containing 2-4 * 10^6 alveolar macrophage cells each. In these wells, 50-60% of the cells had detached from the walls, compared to 0 in non-infected wells. This implies that, for macrophages at least, about 100 infected cells make enough virus to create one plaque in the assay they used. In J Virol. 2015 Jul 1; 89(13): 6773–6781. about 1000 particles per PFU are reported for the 12th passage of virus in their Vero cell assay system, so 10 virus particles per infected cell. We estimated 10^12 virus particles earlier, so that implies about 10^11 cells were infected. That's a similar number as estimated for HIV in the comments of that question you linked. At about 3 * 10^13 human cells in the body, that's about 0.3 % of all cells.
Each of the studies I've linked here is quite different from the other, so plugging results from one into the other is introducing a lot of error, but hopefully the process was informative.