grep -L -Z -r 'Subject: \[SPAM\]' . | xargs -0 -I{} mv {} DIR

The -Z means output with zeros (\0) after the filenames (so spaces are not used as delimeters).

xargs -0

means interpret \0 to be delimiters.

Then

-I{} mv {} DIR

means replace {} with the filenames, so you get mv filenames DIR.