Increased anxiety related to the Coronavirus Disease 2019 (COVID-19) pandemic in society and specific professional groups has been reported by many authors. Most have applied tools enabling assessing the general traits of anxiety. Tools specifically designed for an assessment of anxiety or fear related to COVID-19 have also been developed. However, no study has assessed the future anxiety in relation to the pandemic. This concept was defined by Zaleski in the end of the 20th century as the state of apprehension, fear, worry, and concern regarding unfavourable changes in the more remote personal future. The aim of this study was an analysis to establish the level and the determinants of future anxiety in Polish society related to the COVID-19 pandemic three months after the introduction of the state of epidemic. The analysis reported in the paper is based on the data obtained through a web-based survey carried out on a representative sample of 1002 Polish adults aged 18–74 years. The hierarchical linear regression model was developed for the analysis of the determinants of future anxiety from the responses to a questionnaire consisting of five items. The independent variables selected for inclusion in the model, apart from sociodemographic characteristics, encompassed health literacy (HL) and ehealth literacy (eHL), perceived health threat related to COVID-19 (PHTC19), and a COVID-19-related conspiracy belief score (CCBS) derived from three items asking about the most popular conspiracy theories. The regression model developed in the final step showed that the future anxiety scale score (FASS) was significantly associated with gender, vocational status, HL, PHTC19, and CCBS. The FASS was lower among men than women (regression coefficient (B) (standard error, SE) = −1.28 (0.39), p = 0.001), among entrepreneurs or farmers rather than among employees of the public or private sector (B(SE) = −1.55, p = 0.010), in persons with a higher HL (B(SE) = −0.43 (0.06), p < 0.001). A higher FASS was observed in respondents with higher rather than lower PHTC19 (B(SE) = 1.49 (0.17), p < 0.001) and in those with a higher CCBS (B(SE) = 0.33 (0.07), p < 0.001). The model accounted for 15.2% of the variance of the FASS. In conclusion, the COVID-19 pandemic is not only a cause of increased mental symptoms, but also of increased future anxiety. Health-related measures are significantly associated with the FASS.