A study of over 6000 U.S. Department of Defense personnel, published last year, found that those who had received an annual flu vaccine had higher odds of contracting an infection from the coronavirus family.
Although the flu vaccine was shown to offer significant protection from most influenza viruses and non-influenza coinfections, the odds of contracting coronavirus and human metapneumovirus were significantly higher (OR = 1.36 and 1.51, respectively).
The phenomenon potentially at work here is known as virus interference. It is thought that naturally-acquired influenza infections may reduce the risk of other infections because they temporarily up-regulate innate (non-specific) immune mechanisms.
Worth noting is that other studies have found no association between the influenza vaccine and risk for other infections. However, this study included a greater number of respiratory pathogens that hadn’t previously been studied.