Autophagy improves overall health
A mutitude of Google scholar results confirms that Autophagy (in simple terms, cellular self-digestion) has a myriad of health benefits. The best example is probably this one:
Abstract: Autophagy, or cellular self-digestion, is a cellular pathway involved in protein and organelle degradation, with an astonishing number of connections to human disease and physiology. For example, autophagic dysfunction is associated with cancer, neurodegeneration, microbial infection and ageing. Paradoxically, although autophagy is primarily a protective process for the cell, it can also play a role in cell death. Understanding autophagy may ultimately allow scientists and clinicians to harness this process for the purpose of improving human health. 
Autophagy leads to improved immune function.
This question targets improvements to immune function in particular. This is confirmed by the following articles following articles.
In addition to "simple" breakdown of pathogens, it has also been shown that at least in some cell types (plasmacytoid dendritic cells) autophagy play a role in detection of virus by the so-called pattern recognition receptors (PRR), which are part of the innate immune system. 
The importance of autophagy for cell survival has long been appreciated, but more recently, its essential role in both innate and adaptive immunity has been characterized. Autophagy is now recognized to restrict viral infections and replication of intracellular bacteria and parasites... Thus, the immune system utilizes autophagic degradation of cytoplasmic material, to both restrict intracellular pathogens and regulate adaptive immunity. 
Autophagy plays a key role in the prevention of aging, cancer and neurodegenration but also in the innate immune system against intracellular pathogens. Autophagy has been shown to interact with pattern recognition receptors (PRRs), such as the Toll-like receptors (TLRs). [Ref]
These citations all seem to indicate that, in particular, autophagy of unhealthy cells is a function of good immune response.
*NOTE * My research has yet to show whether autophagy of healthy cells contributes specifically to improved immune function.
Intermittent fasting leads to increased autophagy
Autophagy in the Pathogenesis of Disease (2008) is one study that mentions the effects of fasting in particular:
self-digestion by autophagy—a process that is potently triggered by fasting—is now emerging as a central biological pathway that functions to promote health and longevity. 
This is also confirmed by a 2009 study from Neurobiology of Disease:
Dietary restrictions, including intermittent fasting (IF), provide a non-pharmacological approach to improve the ability of cells to enhance[...] Chaperone production and protein degradation through the autophagy... 
This study focused on Paraneoplastic neurological syndromes, and how intermittent fasting increased autophagy of Schwann cells (which is apparently a good thing).
One final article, a little older than the others, but that will probably interest you more, is a generic treatment of how autophagy is triggered by caloric restriction and leads to a decrease in age-related manifestations that are a result of reduced autophagic activities. 
*NOTE * My research has yet to show whether intermittent fasting leads to increased autophagy of infected cells that would not normally undergo autophagy as part of the natural immune response.
Resources seem to suggest that intermittent fasting is a feasible way to increase autophagy, but by no means suggest that this is the best (most effective or healthiest) method. Many sources suggest that mere caloric restrictions, not outright fasting, are the key. There are some very popular diets that specifically target autophagy. The Atkins Diet relies heavily on ketosis as a method of reducing body fat, which, in itself, could be considered a form of cellular autophagia. [Ref]
It remains to be shown (by my research) whether intermittent fasting promotes autophagy in infected cells where it wouldn't otherwise be triggered, or whether the promotion of autophagy in healthy cells improves to immune function.
 Mizushima, Noboru et al. Autophagy fights disease through cellular self-digestion. February 2008, Nature, 451(7182), pp.1069–1075.
 Lee, Heung Kyu et al. Autophagy-Dependent Viral Recognition by Plasmacytoid Dendritic Cells. March 2007, Science Magazine, 315(5817), pp. 1398-1401.
 Schmid, Dorothee and Münz, Christian. Innate and Adaptive Immunity through Autophagy. July 2007, Immunity 27(1), pp. 11-21.
 Levine, Beth and Kroemer, Guido. Autophagy in the Pathogenesis of Disease. January 2008, Cell, 132(1), pp.27-42,
 Madorsky, Irina et al. Intermittent fasting alleviates the neuropathic phenotype in a mouse model of Charcot–Marie–Tooth disease. April 2009, Neurobiology of Disease,
 Cuervo, AM et al. Autophagy and aging: the importance of maintaining "clean" cells. Dec 2005, Autophagy, 1(3), pp.131-40.