Sharka disease, caused by Plum pox virus (PPV), induces several changes in Prunus. In leaf tissues, the infection may cause oxidative stress and disrupt the photosynthetic process. Moreover, several defense responses can be activated after PPV infection and have been detected at the phytohormonal, transcriptomic, proteomic, and even translatome levels. As proposed in this review, some responses may be systemic and earlier to the onset of symptoms. Nevertheless, these changes are highly dependent among species, variety, sensitivity, and tissue type. In the case of fruit tissues, PPV infection can modify the ripening process, induced by an alteration of the primary metabolism, including sugars and organic acids, and secondary metabolism, including phenolic compounds. Interestingly, metabolomics is an emerging tool to better understand Prunus–PPV interactions mainly in primary and secondary metabolisms. Moreover, through untargeted metabolomics analyses, specific and early candidate biomarkers of PPV infection can be detected. Nevertheless, these candidate biomarkers need to be validated before being selected for a diagnostic or prognosis by targeted analyses. The development of a new method for early detection of PPV-infected trees would be crucial for better management of the outbreak, especially since there is no curative treatment.