Energy Innovation and Carbon Dividend Act of 2023 This bill imposes a fee on the carbon content of fuels, including crude oil, natural gas, coal, or any other product derived from those fuels that will be used so as to emit greenhouse gases into the atmosphere. The fee is imposed on the producers or importers of the fuels and is equal to the greenhouse gas content of the fuel multiplied by the carbon fee rate. The rate begins at $15 per metric ton of CO2-e (i.e., carbon dioxide equivalent) in 2023, increases by $10 each year, and is subject to further adjustments based on the progress in meeting specified emissions reduction targets. The bill also includes exemptions for fuels used for agricultural or nonemitting purposes, exemptions for fuels used by the Armed Forces, rebates for facilities that capture and sequester carbon dioxide, and border adjustment provisions that require certain fees or refunds for carbon-intensive products that are exported or imported. The fees must be deposited into a Carbon Dividend Trust Fund and used for administrative expenses and dividend payments to U.S. citizens or lawful residents. The fees must be decommissioned when emissions levels and monthly dividend payments fall below specified levels. The bill also requires the Department of Energy to enter into agreements with the National Academy of Sciences to study and report on various impacts related to the carbon fee and emissions reduction schedule established by the bill.