This is in no way an "anti-vax" post - it is intended as a serious open discussion as to possible interplay between Influenza vaccine status and non-influenza upper respiratory viruses including Coronavirus (at the least non-novel Coronavirus, but potentially also COVID-19). This is of particular relevance in light of kneejerk government advice further recommending flu shots during the pandemic.
I am not a medical scientist and freely admit limitations in my interpretation of these studies. Influenza vaccines are undoubtedly effective and responsible for saving millions of lives globally, but it is worth recognizing the bias blind spot and ensuring full and transparent analysis of the available evidence in light of the global COVID-19 crisis.
To begin with, these two studies seem to demonstrate potential increased risk of acute respiratory viral infection amongst recipients of influenza vaccine:
The CDC attempts to refute the first of these on their Influenza vaccine misconceptions page as such (bold emphasis mine for discussion below):
There was one study (published in 2012) that suggested that influenza vaccination might make people more susceptible to other respiratory infections. After that study was published, many experts looked into this issue further and conducted additional studies to see if the findings could be replicated. No other studies have found this effect. For example, this article [99 KB, 5 pages] in Clinical Infectious Diseases (published in 2013).
The CDC reference to "many experts" is backed by a single article / study [ref], which itself appears to have several shortcomings, notably:
It examines a completely different and limited cohort compared to other studies, namely children <5 years old and adults ≥50 years old
The results appear to demonstrate a ~1.5% higher rate of non-influenza infections amongst vaccine recipients. This is likely dismissed as statistically insignificant - which is perhaps fair, but equally cannot be defined as an overwhelming debunking of other studies.
Ironically the study itself cites other studies which confirm the effect, thus undermining the CDC's claim that "no other studies have found this effect"
Other studies have found evidence of an association between influenza vaccination and infection with a non-influenza respiratory virus. In 2008, a study of influenza vaccine effectiveness in Australian children 6 to 59 months old found that influenza vaccination was associated with detection of other respiratory viruses 
Further undermining the CDC claim that "no other studies have found this effect" can ironically be found in the following study which was directly funded by the CDC and involved their own researchers:
Vaccinated individuals were no more likely to get influenza after influenza vaccination; however, patients’ experiences of illness after vaccination may be validated by these results which suggest increased risk of ARI caused by non-influenza respiratory pathogens following influenza vaccination among children <18 years.
Note the non-influenza viral pathogens identified by the studies above include Coronavirus strains (excluding of course COVID-19, as they predate it).
Study #3 provides a balanced perspective, again noting the intent is not to undermine Influenza vaccine programmes but rather to ensure potential nuances are well understood:
...the results of the current study should not be interpreted as a reason to change the recommendation for universal vaccination for prevention of influenza, a potentially severe illness with complications such as pneumonia, bacterial infections, hospitalizations, and death 1, , . The influenza vaccine has demonstrated reduction in morbidity and mortality across age groups and baseline health statuses , .
- What is the collective opinion on potential interplay between Influenza vaccination status and risk of contracting acute respiratory viral infections possibly including COVID-19?
- More controversially, is the CDC being overly cavalier in their dismissal of multiple studies (including one of their own) that show such a relationship?
- Is there any data being collected on COVID-19 infections and Influenza vaccination status?