I found a recent paper that says inflammation commonly results in reduced HDL and LDL levels and increases in triglycerides. The reasons are only partially understood.
https://www.ncbi.nlm.nih.gov/books/NBK326741/
The changes in cholesterol levels are apparently a part of the innate immune system response, which is the most ancient part of our immune system.

Here's two quotes from the paper: "The most common changes are decreases in serum HDL and increases in triglycerides. The increase in serum triglycerides is due to both an increase in hepatic VLDL production and secretion and a decrease in the clearance of triglyceride rich lipoproteins. The mechanisms by which inflammation and infection decrease HDL levels are uncertain...LDL levels are frequently decreased but the prevalence of small dense LDL is increased due to exchange of triglycerides from triglyceride rich lipoproteins to LDL followed by triglyceride hydrolysis."

"The changes in lipids and lipoproteins that occur during inflammation and infection are part of the innate immune response and therefore are likely to play an important role in protecting from the detrimental effects of infection and inflammatory stimuli (32,151-153). Some of the potential beneficial effects are listed in Table 3. Thus, the changes in lipid and lipoprotein metabolism that occur during inflammation may initially be protective but if chronic can increase the risk of atherosclerosis.

Table 3.
Beneficial Effects of Lipoproteins

Redistribution of nutrients to immune cells that are important in host defense
Lipoproteins bind endotoxin, lipoteichoic acid, viruses and other biological agents and prevent their toxic effects
Lipoproteins bind urate crystals
Lipoproteins bind and target parasites for destruction
Apolipoproteins neutralize viruses
Apolipoproteins lyse parasites"