Whether Interactive Brokers charges a monthly inactivity fee or not depends on the account type.
IBKR Pro Accounts:
IBKR Pro accounts are subject to a monthly activity fee of USD 10 (USD 20 if the account balance is USD 2,000 or below). If any fee, withdrawal, transfer or adverse market move reduces the account balance to USD 0, the account will be automatically closed.
IBKR Lite Accounts:
IBKR Lite accounts, however, are not subject to a monthly minimum activity fee.