It depends on the version of id3 you are using. For the "id3v2" the tags can be written back to the mp3 directly.
If the mp3 uses "id3v1", the lyrics cannot be written back, because this
version doesn't support the lyrics.

Furthermore, in order to write the lyrics in the tag, the option "Write
lyrics additionally as extra .txt file" must be deselected.

Hope this helps a little