The ongoing Coronavirus (COVID-19) pandemic highlights the inter-connectedness of our present-day globalized world. With social distancing policies in place, virtual communication has become an important source of (mis)information. As increasing number of people rely on social media platforms for news, identifying misinformation and uncovering the nature of online discourse around COVID-19 has emerged as a critical task. To this end, we collected streaming data related to COVID-19 using the Twitter API, starting March 1, 2020. We identified unreliable and misleading contents based on fact-checking sources, and examined the narratives promoted in misinformation tweets, along with the distribution of engagements with these tweets. In addition, we provide examples of the spreading patterns of prominent misinformation tweets. The analysis is presented and updated on a publically accessible dashboard (https://usc-melady.github.io/COVID-19-Tweet-Analysis) to track the nature of online discourse and misinformation about COVID-19 on Twitter from March 1 - June 5, 2020. The dashboard provides a daily list of identified misinformation tweets, along with topics, sentiments, and emerging trends in the COVID-19 Twitter discourse. The dashboard is provided to improve visibility into the nature and quality of information shared online, and provide real-time access to insights and information extracted from the dataset.