With the development of the Internet, social media has become an important channel for posting disaster-related information. Analyzing attitudes hidden in these texts, known as sentiment analysis, is crucial for the government or relief agencies to improve disaster response efficiency, but it has not received sufficient attention. This paper aims to fill this gap by focusing on investigating attitudes towards disaster response and analyzing targeted relief supplies during disaster response. The contributions of this paper are fourfold. First, we propose several machine learning models for classifying public sentiment concerning disaster-related social media data. Second, we create a natural disaster dataset with sentiment labels, which contains nearly 50,00 Twitter data about different natural disasters in the United States (e.g., a tornado in 2011, a hurricane named Sandy in 2012, a series of floods in 2013, a hurricane named Matthew in 2016, a blizzard in 2016, a hurricane named Harvey in 2017, a hurricane named Michael in 2018, a series of wildfires in 2018, and a hurricane named Dorian in 2019). We are making our dataset available to the research community: https://github.com/Dong-UTIL/Natural-Hazards-Twitter-Dataset. It is our hope that our contribution will enable the study of sentiment analysis in disaster response. Third, we focus on extracting public attitudes and analyzing the essential needs (e.g., food, housing, transportation, and medical supplies) for the public during disaster response, instead of merely targeting on studying positive or negative attitudes of the public to natural disasters. Fourth, we conduct this research from two different dimensions for a comprehensive understanding of public opinion on disaster response, since disparate hazards caused by different types of natural disasters.