The question we are considering is if utilitarianism dubs painless killing to be morally impermissible. The simple answer is that no, there is no clear condemnation of painless killing by utilitarian standards. As we know, utilitarianism focuses on two things: the equal value of everyone’s happiness, and the goal of maximizing happiness and minimizing pain. This means that there are any number of extenuating circumstances that would cause a utilitarian to label painless killing as morally permissible or impermissible; for example, she would say it was wrong if the individual she killed had a family to look out for. On the other hand, she would say it was right if killing the individual would benefit many people, for example, killing a serial murderer. However, in the most stripped down example of an individual with virtually no identity, one with no family or friends, and one who does not intend to perform any particularly heroic or heinous actions in the future, utilitarianism provides no reason that killing this individual is wrong. This system of ethics places no particular value on life, independent of its ties to joy and pain. So once these emotions are removed from consideration, the life is worth little to nothing.
The decision of whether or not this act of killing is impermissible depends on the killer. If the killer would suffer grief, guilt, or any emotional distress after performing this action, then it is, by utilitarian standards, morally impermissible. Killing this individual would benefit no one, and hurt the killer. Of course, this consideration is rather illogical, as no one would go through with it under these circumstances. However, on the other hand, if the killer is pathologically deviant, and derives some joy out of killing, then utilitarianism views it as morally permissible, if not obligatory. So for example, suppose I am some strange, sociopathic individual who finds killing for sport (and no other reason) to be enjoyable. If I am presented with a child, one who is orphaned and alone, with no earthly connections and no significant past, utilitarianism states that it is morally permissible for me to kill this child.
It is clear then that, in this specific circumstance, regarding an individual with no connections and no considerable future, utilitarianism states that painlessly killing an individual is morally permissible.