Is there a way to remove the first
N lines from a log that is being actively appended by an application?
No, operating systems like Linux, and it's filesystems, don't make provision for removing data from the start of a file. In other words, the start point of storage for a file is fixed.
Removing lines from the start of a file is usually accomplished by writing the remaining data to a new file and deleting the old. If a program has the old file open for writing, that file's deletion is postponed until the application closes the file.
As commenters noted, for the reasons given in my previous sentence, you usually need to coordinate logfile pruning with the programs that are writing the logs. Exactly how you do this depends on the programs. Some programs will close and reopen their logfiles when you send them a signal (e.g. HUP) and this can be used to prevent log records being written to a 'deleted' logfile, without disrupting service.
There are many utilities available for managing the size of log files, for example logrotate
Some programs have their own utilities. For example, the Apache webserver includes a rotatelogs utility.
I think this task can be achieved with
would remove the lines from 1st to the 10th line form the file.
I think everybody should at least have a look at this sed 1 liners.
Note that this does not work for logfiles that are being actively appended to by an application (as stated in the question).
A safe way to do this would be to halt the application, use sed to truncate the log, then restart the application. This approach can be unacceptable for some services (e.g. a web-server with high throughput and high service-continuity requirements)
No. A solution to this generic problem of log file growth is log rotation. This involves the regular (nightly or weekly, typically) moving of an existing log file to some other file name and starting fresh with an empty log file. After a period the old log files get thrown away.