During the recent Coronavirus disease 2019 (COVID-19) outbreak, the microblogging service Twitter has been widely used to share opinions and reactions to events. Italy was one of the first European countries to be severely affected by the outbreak and to establish lockdown and stay-at-home orders, potentially leading to country reputation damage. We resort to sentiment analysis to investigate changes in opinions about Italy reported on Twitter before and after the COVID-19 outbreak. Using different lexicons-based methods, we find a breakpoint corresponding to the date of the first established case of COVID-19 in Italy that causes a relevant change in sentiment scores used as proxy of the country reputation. Next, we demonstrate that sentiment scores about Italy are strongly associated with the levels of the FTSE-MIB index, the Italian Stock Exchange main index, as they serve as early detection signals of changes in the values of FTSE-MIB. Finally, we make a content-based classification of tweets into positive and negative and use two machine learning classifiers to validate the assigned polarity of tweets posted before and after the outbreak.