In short: There is no evidence to claim that high intake of refined (or other) sugars impairs the digestion of proteins, but may stimulate inflammation via other mechanisms.
The most commonly used refined (added) sugars (glucose or dextrose, fructose, HFCS, invert sugar, corn syrup) are monosaccharides, which do not need to be digested, so they do not require or "deplete" any digestive enzymes; sucrose is digested by the enzyme sucrase and maltose by the enzyme maltase (Medical LibreTexts).
Proteins are digested by other enzymes that have nothing with the digestion of sugars: pepsin, trypsin, chymotripsin and peptidases, and whole or "partially digested proteins" are, in general, not absorbed (VIVO Pathophysiology).
Partially digested proteins and certain toxins may be absorbed in "leaky gut syndrome," a pathological condition with increased intestinal permeability, but there seems to be no evidence that sugars would contribute to it.
Dietary sugar may increase inflammation:
There is some evidence that high intake of refined sugars may increase the risk of inflammatory bowel disease (IBD):
high pre-illness (IBD) intake of refined sugars and low fiber has
been observed in numerous studies [4, 21, 43] but two large
observational cohorts found no association between baseline sugar
intake and IBD [22, 44] (Indian Journal of Gastroenterology, 2018).
It has been postulated that dietary sugar consumption contributes to
increased inflammatory processes in humans. Central to the potentially
relevant mechanisms is the fact that dietary sugar promotes de novo
synthesis of free fatty acids (FFA) in the liver [17,18,19], which
according to the lipotoxicity theory, would produce FFA metabolites
that may trigger inflammatory processes and reactive oxygen species
(ROS) formation [20,21] (Nutrients, 2018)
Diet rich in saturated fat, trans-fats, or refined sugar is associated
with higher production of pro-inflammatory molecules, especially in
individuals with diabetes or overweight individuals (StatPearls, 2019).