Scenario

You have a MySQL Master-Master replication setup, Lets say Master1 (LIVE) and Master2 (DR Warm Standby)
Master2 server has been restarted for some reason.
Now Master1 LIVE (the slave of DR) slave status is stopped with the below error.

Last_IO_Errno: 1236 Last_IO_Error: Got fatal error 1236 from master when reading data from binary log: 'Could not find first log file name in binary log index file'

Solution

For the safer side, Backup your database used by the application with mysqldump and keep it in a safe place. The command usually wont cause any data corruption on downtime.

Master1: stop slave;

Master2: flush logs;
Master2: show master status; — take note of the master log file and master log position

mysql> show master status;
+-------------------+----------+--------------+------------------+
| File | Position | Binlog_Do_DB | Binlog_Ignore_DB |
+-------------------+----------+--------------+------------------+
| master-bin.000006 | 106 | | |
+-------------------+----------+--------------+------------------+

Master1: CHANGE MASTER TO MASTER_LOG_FILE = 'master-bin.000006', MASTER_LOG_POS = 106;
Master1: start slave;

Now run show slave status \G; and verify everything is up and running.