What you want to do, can definitely be done with FTP. Technically it's the same, what any FTP client does, when resuming an interrupted file download.
Though from a user perspective, I do not know, if any FTP client supports an explicit download of only given number of trailing bytes.
But some FTP clients will definitely allow you to download new trailing contents of file that you have previously downloaded.
Particularly with WinSCP, just initiate a log file download. Then, on the Overwrite confirmation prompt, select the Resume (it's in drop down menu of the No button). Note that the option is obviously available, only if the source file is larger than the destination file.
If you really want to download only the last few lines of the log, you can cheat WinSCP, by creating a dummy local file with a size bit smaller than the log file before you initiate download.
You can also easily automate the above trick:
fsutil file createnew mylog.log 100000000 winscp.com /command "open mysession" "get -resume /path_to_log/mylog.log" "exit"
For alternatives to the fsutil
, see Quickly create large file on a windows system?
With a more effort you can modify the script to first check log file size and automatically calculate dummy file size few kilobytes smaller than the log size.