Q:

Is it more harmful to eat sugary food when sick?

If you search junk food when sick there's all sorts of info telling you not to eat it, e.g.
Natural Society: 5 Foods NOT to Eat While Sick or Battling a Cold

5 Foods NOT to Eat While Sick or Battling a Cold
#5 Fast Food and Processed Foods
Surprise, surprise – it should be most obvious to steer clear of fast food and processed foods while sick. Fast food and most processed foods are void of nutritional value; that is, they bring hardly anything to the table in terms of boosting the immune system or amplifying your health. Not only are much of these foods made from genetically modified foods, but they also contain a slew of immune-suppressing ingredients such as high fructose corn syrup, aspartame, MSG, artificial colors, FD&C Red #40, dimethylpolysiloxane, and much more. And despite some irresponsible hospitals serving up McDonald’s to their sick patients, that doesn’t mean these foods are any more valuable.

This surprises me, as I thought (1) the easy available energy in sugary junk food would assist in battling the infection, (2) I don't believe GMO foods are intrinsically bad for you, and (3) any alleged immune-suppressing ingredients would be long-term issues, not short term.
Is there evidence that eating sugary foods when sick is a particularly bad time to eat them?

A:

No, unless you are obese.
According to the National Institute of Health1:

In conclusion, when considering the signaling pathways involved in glucose metabolism in immune cells, it is generally accepted that glucose uptake and metabolism are promoted by PI3-K and its downstream signaling molecule Akt (both in T- and B-lymphocytes). mTOR appears to be more critical in favoring efficient protein translation and inhibiting protein degradation.

In some simpler terms, increased sugar (glucose) increased T Cell (immune cells) response. That may put a whole new outlook on honey being used as a medical supplement.

Now, that all gets flipped onto its head when someone is obese. In the exact same research paper:

As in human obesity, obese animals present a delayed wound healing associated with increased polymorphonuclear cell infiltration. In addition, both T- and B-cell-mediated immune responses were reported to be impaired in obese ob/ob and diabetic db/db mice.

What would obesity have to do with it? They explain:

Leptin is an adipocyte-derived cytokine, secreted proportionally to the amount of fat, originally characterized for its capacity to finely regulate body weight. Indeed, the complete congenital absence of leptin leads to a syndrome of intense hyperphagia and morbid obesity both in humans and rodents, which can be reverted by administration of the recombinant molecule. Interestingly, subsequent studies further demonstrated that leptin intervenes in both innate and adaptive immunities. Leptin promotes activation of monocytes/macrophages chemotaxis and activation of PMN cells, development and activation of natural killer (NK) cells, and regulation of T cell responsiveness.

In simpler terms, obese people become deficient in Leptin, which is directly related to the responsiveness of T Cells.

Conclusion in simpler terms

A healthy person, with no metabolic disorders (like obesity) could
actually benefit from having sugar in their diet, because it would
increase the reaction of their body's immune system.
An unhealthy person, with a metabolic disorder (like obesity) would
not benefit from having sugar in their diet, because it would
decrease the reaction of their body's immune system.

1Feeding Our Immune System: Impact on Metabolism
(Wolowczuk, Isabelle et al. “Feeding Our Immune System: Impact on Metabolism.” Clinical and Developmental Immunology 2008 (2008): 639803. PMC. Web. 9 Nov. 2015.)

