D&D 3.5e
In the original, D&D 3.5e Eberron Campaign Setting book, the living spell template is given this description:
“Living spell” is an unusual template, in that it is applied to an arcane or divine spell effect (or in some cases, a group of spell effects) and not to a creature. The characteristics of a living spell are determined by the nature of the spell(s), including the caster level of the spells. The template can be applied to any spells that create an area or effect (not targeted spells), but not a spell whose effect is already a creature (such as a summon monster spell).
Since the cure X wounds spells of 3.5e are targeted spells, they would not be eligible for becoming living spells.
However, there is no rule against “friendly” living spells. If some spell created an area of healing, for example, it could become a living spell. I’m not aware of any such spells that provide healing, however—though 3.5e had mass versions of many healing spells, these were multi-target, not area-effect. The only healing area spell I can find is Player’s Handbook II’s righteous burst—which healed allies while harming enemies, and so presumably does the same to the living spell’s allies and enemies.
There is, however, a spell that “create[s] an [...] effect,” and that effect applies healing—Player’s Handbook II’s healing spirit, which conjures a “spirit” of light that heals creatures by touching them. Healing spirit is eligible for the living spell template—technically. The spirit isn’t defined as a “creature” and doesn’t have stats like a creature, so RAW, it doesn’t fall into living spell’s prohibition on effects-that-are-creatures. But it should probably be nixed for the same reasons that summon monster is—it’s not at all clear to me how a living healing spirit would actually work. For example, the template says “A creature hit by a living spell’s slam attack is subjected to the normal effect of the spell [...] making up the creature.” No one is really “subjected” to healing spirit directly, the spell just conjures a spirit that you can direct to heal people afterwards. I suppose every slam attack conjures a new spirit? The target of the slam isn’t really “subjected” to it, though. The living spell’s engulf attack isn’t much better—I guess those conjure a new spirit each round it’s engulfing something? Pretty ridiculous. Moreover, the conjured spirits probably disappear when the attack and/or engulfing are over, which is weird too.
Anyway, there was, for the record, a prestige class in 3.5e called the spell sovereign, whose signature feature was a living spell companion. Various ways of controlling oozes—living spells are considered an ooze at least in 3.5e—could also get a friendly living spell. In the case of righteous burst, once friendly to the party it would presumably heal them.
D&D 5e
The 5e description of the living spell from Eberron: Rising from the Last War restricts it to conjurations or evocations for no good reason I can imagine (I’m pretty sure 3.5e adventures had living spells of other schools, and I know there have been other schools in games I’ve played and run), but does not require that it only be applied to an area or effect spell, so there is no problem with a single-target spell like cure wounds. And since cure wounds is an evocation in 5e (it was a conjuration in 3.5e, and has been a necromancy in other editions), it’s legal for becoming a living spell.
However, Eberron: Rising from the Last War does instruct DMs to take a spell “that deals damage” when creating a living spell. The way this is phrased is as an instruction, not a requirement, which is very strange. Whether you want to treat it as a requirement or not is, I suppose, up to you.
Finally, for the record, there is a 5e version of healing spirit, which works similarly and raises the same questions as the 3.5e version. It’s in Xanathar’s Guide to Everything. Thanks to Korvin Starmast for pointing it out, which also led me to the 3.5e version.
