If it is possible to you, you should replace the calls to dbms_output.put_line by your own function. 
Here is the code for this function WRITE_LOG  -- if you want to have the ability to choose between 2 logging solutions:
CREATE OR REPLACE PROCEDURE to_dbg_table(p_log varchar2)
  -- table mode: 
  -- requires
  -- CREATE TABLE dbg (u varchar2(200)   --- username
  --                 , d timestamp       --- date
  --                 , l varchar2(4000)  --- log 
  -- );
AS
   pragma autonomous_transaction;
BEGIN
  insert into dbg(u, d, l) values (user, sysdate, p_log);
  commit;
END to_dbg_table;
/
or write directly to the DB server that hosts your database
This uses the Oracle directory TMP_DIR
CREATE OR REPLACE PROCEDURE to_dbg_file(p_fname varchar2, p_log varchar2)
  -- file mode: 
  -- requires
--- CREATE OR REPLACE DIRECTORY TMP_DIR as '/directory/where/oracle/can/write/on/DB_server/';
AS
  l_file utl_file.file_type;
BEGIN
  l_file := utl_file.fopen('TMP_DIR', p_fname, 'A');
  utl_file.put_line(l_file, p_log);
  utl_file.fflush(l_file);
  utl_file.fclose(l_file);
END to_dbg_file;
/
WRITE_LOG
Then the WRITE_LOG procedure which can switch between the 2 uses, or be deactivated to avoid performances loss (g_DEBUG:=FALSE).
CREATE OR REPLACE PROCEDURE write_log(p_log varchar2) AS
  -- g_DEBUG can be set as a package variable defaulted to FALSE
  -- then change it when debugging is required
  g_DEBUG boolean := true;
  -- the log file name can be set with several methods...
  g_logfname varchar2(32767) := 'my_output.log';
  -- choose between 2 logging solutions:
  -- file mode: 
  g_TYPE varchar2(7):= 'file';
  -- table mode: 
  --g_TYPE varchar2(7):= 'table';
  -----------------------------------------------------------------
BEGIN
  if g_DEBUG then
    if g_TYPE='file' then
      to_dbg_file(g_logfname, p_log);
    elsif g_TYPE='table' then
      to_dbg_table(p_log);
    end if;
  end if;  
END write_log;
/
And here is how to test the above:
1) Launch this (file mode) from your SQLPLUS:
BEGIN
  write_log('this is a test');
  for i in 1..100 loop
    DBMS_LOCK.sleep(1);
    write_log('iter=' || i);
  end loop;
  write_log('test complete');
END;
/
2) on the database server, open a shell and 
    tail -f -n500 /directory/where/oracle/can/write/on/DB_server/my_output.log