暖男 means men who are considerate and care very much about the girls' feelings, but are sometimes too willing.
Do girls nowadays like them? It really depends on the girl and opinions vary from person to person. As far as I can see, they are certainly genuine and decent good men but at the same time they are also certainly not the popular type, in my opinion mainly because they are too willing and care too little about themselves. Like puppies, only end up getting used and taken for granted. Women who like them are either club trash who are tired of being deceived and could use a rebound date, or miserable over paranoid virgins who never bothered taking a look into the mirror yet daydream everyday about encountering a perfect man till they grow old.