Xanathar's now limits healing spirit heavily
The spell now includes the following (a tweet of the book agrees with this, as does this video, and this confirmation from Jeremy Crawford that the spell received an update):
The spirit can heal a number of time equal to 1 + your spellcasting ability modifier (minimum of twice). After healing that number of times, the spirit disappears.
This means that the number of instances of healing is limited directly by the caster's spellcasting ability modifier which, as far as I'm aware, has no way of getting above 10, thus putting the absolute upper limit at 11.
The following part of the answer addresses the functioning of the spell without this limitation:
Each person can heal at least twice per round for a total of 80 heals
This entire gimmick depends on the Ready action:
[...] First, you decide what perceivable circumstance will trigger your reaction. Then, you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it [...]
Using this you could walk through the healing spirit, and then use the Ready action to move on somebody else's turn, thus activating another instance of healing. The process would be as follows:
- The caster casts healing spirit and walks into it (1 heal)
- The caster takes the Ready action to move after somebody else goes through the healing spirit
- The next person (Bob) walks through it, triggering the caster's Readied action (1 heal)
- The caster walks through the healing spirit (1 heal)
- Bob takes the Ready action to move after somebody else goes through the healing spirit.
- And so forth until the spell ends.
This gives a total of 2 healings per person per round. The spell lasts for 10 rounds and since there are 4 party members this adds up to 80 instances of healing over the course of the 1 minute spell.
I could not find any way to move through healing spirit on somebody else's turn that does not expend a creature's single reaction/round.
It might be possible to do even more
The spell states:
[...] Whenever you or a creature you can see moves into the spirit's space for the first time on a turn or starts its turn there [it heals]
This could be read as meaning that if a creature starts its turn in the healing spirit, it heals, and then, if it exits and re-enters the healing spirit it heals yet again, even though it's still the same turn. This isn't a natural reading and it would also apply to all the other spells with similar wording and it also isn't a reading I would ever use. But if you really wanted to you could.
Using this things get a bit more complicated but it would now be possible for everybody to heal three times per round (except some people on their first turn).
Let's have the party consist of Alice the caster, Bob and Charlie for simplicity (the proof for a party of 4 is left to the reader). The plan would be as follows:
Round 1:
- Alice casts healing spirit on Bob and walks through it (heal).
- Alice takes the Ready action to move onto the healing spirit after Charlie moves through it.
- Bob is standing on the healing spirit (heal)
- Bob walks out of and back through the healing spirit (heal)
- Bob takes the Ready action to move onto the healing spirit after Alice moves through it.
- Charlie moves through healing spirit (heal)
- Alice's Readied action triggers and she walks onto the healing spirit before her turn begins. (heal)
- Charlie takes the Ready action to move onto the healing spirit after Bob moves through it.
Round 2:
- Alice is standing on the healing spirit (heal)
- Alice walks out of and back through the healing spirit (heal)
- Bob's Readied action triggers and he enters the healing spirit (heal)
- Alice takes the Ready action to move onto the healing spirit when Charlie moves through it.
- Bob is standing on the healing spirit (heal)
- Bob walks out of and back through the healing spirit (heal)
- Charlie's Readied action triggers and he enter the healing spirit (heal)
- Bob takes the Ready action to move onto the healing spirit after Alice moves through it.
- Charlie is standing on the healing spirit (heal)
- Charlie walks out of and back through the healing spirit (heal)
- Alice's Readied action triggers and she walks onto the healing spirit before her turn begins. (heal)
- Charlie takes the Ready action to move onto the healing spirit after Bob moves through it.
Round 3:
- Alice is standing on the healing spirit (heal)
- Repeat round 2.
That's a total of 5 heals in the first round but 9 in the second. With a party of four there would be 6 heals in the first round and 12 on any other. This gives a total of 114 heals over the course of 1 minute.