Since its outbreak, the ongoing COVID-19 pandemic has caused unprecedented losses to human lives and economies around the world. As of 18th July 2020, the World Health Organization (WHO) has reported more than 13 million confirmed cases including close to 600,000 deaths across 216 countries and territories. Despite several government measures, India has gradually moved up the ranks to become the third worst-hit nation by the pandemic after the US and Brazil, thus causing widespread anxiety and fear among her citizens. As majority of the world's population continues to remain confined to their homes, more and more people have started relying on social media platforms such as Twitter for expressing their feelings and attitudes towards various aspects of the pandemic. With rising concerns of mental well-being, it becomes imperative to analyze the dynamics of public affect in order to anticipate any potential threats and take precautionary measures. Since affective states of human mind are more nuanced than meager binary sentiments, here we propose a deep learning-based system to identify people's emotions from their tweets. We achieve competitive results on two benchmark datasets for multi-label emotion classification. We then use our system to analyze the evolution of emotional responses among Indians as the pandemic continues to spread its wings. We also study the development of salient factors contributing towards the changes in attitudes over time. Finally, we discuss directions to further improve our work and hope that our analysis can aid in better public health monitoring.