If you can't kill your application, you can truncate instead of deleting the log file to reclaim the space. If the file was not open in append mode (with O_APPEND), then the file will appear as big as before the next time the application writes to it (though with the leading part sparse and looking as if it contained NUL bytes), but the space will have been reclaimed.
To truncate it: #> /path/to/the/file.log
If it was already deleted, on Linux, you can still truncate it by doing:
#> "/proc/$pid/fd/$fd"
Where $pid is the process id of the process that has the file opened, and $fd one file descriptor it has it opened under (which you can check with lsof -p "$pid".
If you don't know the pid, and are looking for deleted files, you can do:
#lsof -nP | grep '(deleted)'
Or (on Linux):
#find /proc/*/fd -ls | grep '(deleted)'
Or to find the large ones with zsh:
ls -ld /proc/*/fd/*(-.LM+1) | grep '(deleted)'
An alternative, if the application is dynamically linked is to attach a debugger to it and make it call close(fd) followed by a new open("the-file", ....).
