The official definition of demure is: "reserved, modest, and shy."
But does it also imply submissiveness?
Demure used on its own doesn't necessarily imply submissive. Dictionary.com fails to list submissive as a direct synonym, although it does cite words such as timid, unassertive and constrained. Demure can be used by itself to imply a certain kind of affect that is far from submissive. Wordnet links demure with coy and overmodest (affectedly modest or shy especially in a playful or provocative way).
Authors are inclined to use both demure and submissive together to convey a mix of traits. Wordnik, noted above, cites this passage from Never Come Down: "'Demure' and 'submissive' were not words in her vocabulary, thus making her very nearly the opposite of the feminine ideal of her day.” Lady Chatterley's Lover has this passage: "A man of her own class he would not mind, for Connie was gifted from nature with this appearance of demure, submissive maidenliness, and perhaps it was part of her nature."
I think the OED does a really good job of explaining the adjective "demure." I chose the most illustrative examples.
I don't think demure in sense 2 has any negative connotations. However, demure used in sense 3 clearly suggests unnaturalness.
2.Of persons (and their bearing, speech, etc.): Sober, grave, serious; reserved or composed in demeanour.
1523 J. Skelton Goodly Garlande of Laurell 902 Demure Diana, womanly and sad.
3.Affectedly or constrainedly grave or decorous; serious, reserved, or coy in a way that is not natural to the person or to one of his years or condition [emphasis mine - Alex B.]
a1692 T. Shadwell Volunteers (1693) iii. i. 25 This Gentleman, and his demure Psalm-singing Fellows.
I don't think that the idea of submissiveness is present in demure.
A submissive person is generally obedient and does what they're told to do.
A demure lady could be reserved, modest and shy but still say no - and in the days when ladies were more demure than they are now, would have been expected to do so for certain situations!