SQL Error log maintenance
Are there any performance gains for recycling the SQL Error log?


Not in terms of sql server's performance. Better for maintenance and when you want to find specific error - the size of the log file will affect any app or even opening the error log.

To recycle error log - use EXEC sp_cycle_errorlog. You can even schedule this using a sql agent job on a weekly or biweekly (depending on the activity of your server).

If you have Logshipping, to avoid getting your log bloat, I use TF 3226


  How often should you recycle it?


It depends ! I have it every week on my prod servers with a maximum of 10 error logs (Limit the number of error log files before they are recycled setting in sql server log --> configure option) .

This is official from Microsoft - How to manage the SQL Server error log ?