How to stop SQL Server from logging informational lines to SQLFT log files?

If you can't restart the instance, you can schedule a log cleanup with sp_fulltext_recycle_crawl_log. More info: http://www.ruben-j.com/microsoft/t-sql/sql-fulltext-crawl-log/ or https://social.msdn.microsoft.com/Forums/windowsapps/en-US/b9420843-666e-4f33-b987-2624cdfa7acd/sql-server-2008r2-fulltext-logfile-sqlftlog-maintenance?forum=sqlkjmanageability or post the question to dba.stackexchange.

In the end the solution was to add to Task Scheduler a simple one-liner powershell script which will clear the SQL Server logs older than 30 days, executed every day:

Get-ChildItem "C:\Program Files\Microsoft SQL Server\MSSQL11.MSSQLSERVER\MSSQL\Log\*" -Recurse | Where-Object {$_.LastWriteTime -lt (Get-Date).AddDays(-30)} | Remove-Item

Change MSSQL11.MSSQLSERVER to match your SQL Server version and instance name.