Yes, it can heal the target. (Qualified)
There are a variety of healing (as in hit point recovery) mechanisms in the PHB that do not require (a) some sort of medical knowledge/skill, or (b) magic, including spending hit dice during a short rest, hit point recovery during a long rest, temporary hit points accruing to a character and/or her allies as a result of (non-magical) class features. Indeed, in the real world, the field of psychosomatic medicine is concerned with precisely the question of how one's beliefs, perceptions, emotions and state of mind influence health and healing (e.g., placebo effects are one such example).
Therefore it is reasonable to infer that phantasmal force can cause healing (temporary or permanent). Of course, 5th ed. D&D is predicated on rules in the broad stroke, with DM fiat, rulings or house rules on the details (for example, such healing might be ruled to require the availability and expenditure of hit dice, etc.), so of course "it's up to the DM."
One might argue that the psychic damage mechanic described in the write up for phantasmal force somehow indicate only a possibility of psychic harm. I disagree that there is no mechanism for healing (see my first paragraph) given the spell description's emphasis on effects on the beliefs of the target creature.