I think the easiest way to do is with backticks
mv `ls -1 ~/Linux/Old/ | grep -v Tux.png` ~/Linux/New/
Edit:
Use backslash with ls instead to prevent using it with alias, i.e. mostly ls is aliased as ls --color.
mv `\ls -1 ~/Linux/Old/ | grep -v Tux.png` ~/Linux/New/
Thanks @Arnold Roa