The act of tacitly encouraging a mentally-ill, self-harming person to continue self-harm does not count as free speech. The reason is that when Person X transmits speech to Person Y, whether that transmission counts as free speech must hinge upon Person Y (a) consenting and (b) having the contractual capacity to consent in the first place. For example, if someone called your telephone landline a hundred times every day without your permission, that would not be free speech, as you have withheld consent to receive the message. You are able to offer or withhold consent because you retain contractual capacity.
When someone repeatedly and visibly self-harms, that person indicates that she or he is contractually incompetent in some important respects. For example, suppose there is plausible reason to believe an able-bodied, physically healthy young man is on the verge of suicide, and his girlfriend tells him "Go ahead and do it!", and then he does it. That man lacked the contractual capacity to judge for himself whether that was sound advice, and therefore he was not in a position to consent to it. It is comparable to having a person sign his estate over to you when he is drunk.
If you see an angry crowd and tell the crowd "Go beat up that Person, T, over there!” and then the crowd does beat up Person T, that is an incitement to violence and it counts as an initiation of the use of force. Likewise, telling an able-bodied, physically healthy, suicidal person to kill himself is to incite a person to perform violence upon himself when was not in a position to offer or withhold genuine consent.
There are actually websites and Facebook groups that advocate anorexia as if it were a legitimate lifestyle choice. This is not free speech. The reason is that, qua anorexic person, the person suffering from anorexia is not contractually competent enough to offer genuine consent to the pro-anorexia message.
One might counter that it is unfair to say that anorexics are contractually incompetent in this context, as they appear to hold capacity in most contexts where contractual capacity is necessary. For example, anorexics often hold normal jobs and make long-term, high-stakes financial decisions such as going to university and purchasing cars or homes or parking garages. If such people are not competent to consent to receiving pro-anorexia messages, then would that not mean that they are not competent to make these other big decisions -- decisions that involve long-term commitment and come with great financial risk -- that require contractual capacity? The answer is that contractual capacity can be context-based. You can be fairly considered contractually competent most of the time but contractually incompetent when you are drunk. Likewise, it can be fairly stated that an anorexic person is contractually competent in most contexts but should still not be considered contractually competent as far as this eating disorder is concerned.
Now, I must address this frequent equivocation: "Obesity is unhealthy too. If you think that pro-anorexic speech should not be called free speech, then the same standard must apply to any expression that might promote obesity." That is an absurd conflation. Anorexia is physically dangerous in a way that obesity is not. Leaving aside the morbidly obese (who also have a psychological issue), someone who has been overweight since adolescence -- and has remained obese consistently -- can often still live into his or her sixties. By contrast, someone who is consistently anorexic in her teens seldom lives to reach age 40. A twenty-year-old who is overweight has several decades to change her lifestyle; a twenty-year-old anorexic does not. There is a greater physiological urgency in addressing anorexia, and therefore people should stop conflating the two conditions. (For information on the greater urgency of addressing anorexia, see this, this, this, this, this, and this.)
John P. McCaskey points out that contract breach and fraud are both forms of the initiation of physical force wherein the perpetrator manipulates the victim into taking the physical action that harms the victim. The same logic applies to reinforcing and encouraging the morbid gestures of someone who has a history of self-harm. Insofar as self-harm is concerned, the self-harmer is not contractually competent to evaluate the merits of someone's encouragement or reinforcement of the self-harm. Thus, reinforcement and/or encouragement of the morbid gestures is manipulation of the victim no less harmful than is fraud.
Some years ago (the same year I started this blog), I knew someone who had a history of threatening suicide, and who later began uploading disturbing pictures of herself wherein she was photoshopped as a corpse. Those morbid gestures were clearly related to the years of suicidal ideation. Upon making this context known to the group of Norwegians encouraging the uploading of the corpse imagery and other morbid gestures, those Norwegians brushed off such concerns; the morbid gestures went on being reinforced. Those reinforcers of those suicidal gestures were engaging not in free speech but contributing to someone's self-harm.