As this question is somewhat extensive, I have divided the post into headers in order to make it easier to read and follow.
Charlie Gard is a British baby who has a rare genetic disorder called MDDS, leaving him blind, deaf, and unable to breathe without outside support. He's terminally ill, but there is an experimental treatment available in the US that could save his life.
The problem is that the hospital refuses to let him go, saying that prolonging his life will cause undue suffering; the best thing, in their opinion, is to take him off of life support. The parents sued the hospital, but they lost and were denied appeal. This has sparked an international controversy over whether the parents should be allowed to bring their child to the United States for treatment.
The Mishnah in Oholos (7:6) states:
האשה שהיא מקשה לילד, מחתכין את הולד במעיה ומוציאין אותו אברים אברים, מפני שחייה קודמין לחייו. יצא רבו, אין נוגעין בו, שאין דוחין נפש מפני נפש.
[If] a woman is having a difficult time in labor, we may kill1 the fetus inside her and bring it out limb by limb, for her life precedes his. [If] his majority2 has exited, we may not touch him, for we do not push aside one life in favor of another.3
As Rashi to Sanhedrin 72a explains:
דכל זמן שלא יצא לאויר העולם לאו נפש הוא וניתן להורגו ולהציל את אמו אבל יצא ראשו אין נוגעים בו להורגו דהוה ליה כילוד ואין דוחין נפש מפני נפש
For however long he has not exited into the air in the world, he is not considered alive, and he may be given to be killed to save his mother. However, "[if] his head has exited, we may not touch him" to kill him, for he is considered as having been born, and "we do not push aside one life in favor of another."
The reason I bring down these sources is to prove that, barring any extenuating circumstances (see footnote 3), one may not kill a baby; even at birth, it is considered alive, and certainly once it passes the threshold of thirty days.4,5
Perhaps another thing to consider is that murder is an exception to the general rule that messengers don't obligate in their senders for sins performed in their service (ein shliach l'dvar aveirah) - that is, if I hire a mercenary, I'm liable for his murdering (Kiddushin 43a, Rambam Hilchos Retzichah 2:2-3). Even though the government is not Jewish, even non-Jews are halachically sheluchim if they could perform the agency on their own (Kiddushin 41b); as non-Jews are commanded in murder as well, this would seem to qualify.
Let's return to the Charlie Gard case, and let's imagine if the family was Jewish. As we discussed above, a baby is unable to be killed once it's born; it has the status of being alive, just like anyone else.
The logical extension of that is that when a baby needs to be put on life support, it should be forbidden to unplug it, just as it is for anyone else - if it's murder for the elderly, it should be murder for the baby as well, because, again, we saw above that we make no distinction in this regard.
So what happens when the hospitals and courts refuse to allow treatment to a baby? At what point are the parents allowed - or, perhaps, obligated - to give up? To what extent must they fight the system before being satisfied knowing they did everything that they could? By consenting to the government putting the child down, are the liable for the government's murder, as per the latter discussion above in the sources?6
1Literally translated, the word מחתחין means to cut. I have translated it as to kill based on the Rambam in Hilchos Rotzeiach 1:9 that one may kill the fetus in any manner in order to save the mother, based on the halacha of rodef. See note 3.
2The version in Sanhedrin 72a has "his head" rather than "his majority." For our purposes, this does not make a difference.
3The Gemara in Sanhedrin ibid. notes that this seems to go against the halachos of rodef, but answers that משמיא קרדפי לה, literally translated as "from Heaven she is being chased." The meaning of this cryptic line is beyond the scope of this discussion; the interested reader may refer to Rashi and Meiri ad. loc., and Rambam ibid.
4See, for instance, Rambam, Hilchos Milah 1:13, where he discusses that a baby that lives for 30 days is considered viable.
5The reason I discuss that it's considered murder once the baby is born, rather than discuss that abortion is forbidden, is because it's unclear what exactly the prohibition is; with the exception of R' Moshe on that list, nobody holds it's murder, and even R' Moshe is uncertain whether it is considered full-fledged murder. To avoid this sticky case, I decided to stick with the sources that are more solid to set up my question.
6The reason I ask about a case with an infant rather than an older patient is because once a patient is 18 (in both US and UK law) the hospitals address such issues with the patient directly, as he or she is legally mature enough to make such decisions on their own. Even in the case of younger patients (I can only surmise based on my own medical experience) doctors will often discuss with the child what their wishes are, even if they decide it's in their best interest not to abide. However, with an infant, even that option is off the table; the parents are the only ones able to defend the child.