Hundreds of millions of people worldwide are affected by viral infections each year, and yet, several of them neither have vaccines nor effective treatment during and post-infection. This challenge has been highlighted by the COVID-19 pandemic, showing how viruses can quickly spread and how they can impact society as a whole. Novel techniques that bring in different disciplines must emerge to provide forward-looking strategies to combat viral infections, as well as possible future pandemics. In the past decade, an interdisciplinary area involving bioengineering, nanotechnology and information and communication technology (ICT) has been developing, known as Molecular Communications. This new emerging area uses elements of classical communication systems and maps it to molecular signalling and communication found inside and outside the body, where the aim is to develop new tools that can serve future medicine. In this paper, we provide an extensive and detailed discussion on how Molecular Communications can be integrated into the research on viral infectious diseases modelling, and how possible treatment and vaccines can be developed considering molecules as information carriers. We provide a literature review on the existing models of Molecular Communications for viral infection (in-body and out-body), a deep analysis on their effects on the host and subsequent communication process for other systems within the body (e.g., immune response), sources of experimental data on known viral infections and how it can be used by the Molecular Communications community, as well as open issues and future directions. Since the development of therapeutics/vaccines needs an interdisciplinary approach centred around ICT, we are confident that Molecular Communications can play a central role here by providing a detail characterisation and manipulation of the propagation of molecules in different media.