I study the dynamic implications of climate policy risk on macroeconomic outcomes and asset prices. I find that accounting for climate policy that restricts oil use and has an unknown arrival time, in an otherwise standard climate-economic model with oil extraction, generates a run on oil; meaning oil firms dynamically accelerate oil extraction as climate change increases and oil reserves decrease due to the risk of future climate policy actions stranding oil reserves. Furthermore, the risk and uncertainty of the climate policy, and the run on oil it causes, leads to a downward shift and dynamic decrease in the oil spot price and value of oil firms compared to the setting without climate policy risk. Empirical evidence based on cross-sectional and time series analysis demonstrates that the effects of climate policy risk on observable market outcomes such as oil production, stock returns, and oil prices are consistent with the predictions of the model.