What is a common idiom meaning, 'that a recovering patient's medical situation is likely to change all of a sudden without notice or reason'?

One phrase that fits well for this situation is touch and go - it means that the outcome of a situation is uncertain, and that things can change at a moment's notice. A person in the hospital after a severe accident may be in "touch and go condition" if it's unsure if they'll survive, and it can be used to highlight the unpredictability of the outcome even if it seems they are getting better. A situation that is "touch and go" can change rapidly and dramatically.

When things can change course rapidly, one expression is "to turn on a dime." In general, the phrase "Life can turn on a dime" or "Life turns on a dime" applies to the big picture to include appreciating what you have and how things can change at any moment. Change can go in different directions, so it's not necessarily a foreboding phrase as much as it marks an abrupt switch. You could say someone's behavior turns on a dime when the changes are sudden. (TFD)

Medical parlance when talking about someone whose condition is not getting any worse is, "stable." You often hear this in news reports after someone is injured, "Doctors describe the victim's condition as critical but stable."

While it would probably offend a doctor's sensibilities to describe a patient's condition as, "unstable," from someone who wasn't a doctor the meaning would be clear.

Unstable
d(1): liable to change or alteration
MW

A medical professional would probably use the phrase, "Their condition is not yet stable," or, "We are trying to stabilise them," mostly to avoid the implications of the word unstable and it's association with mental health.

I do like the other answers as well. I guess in such a situation, you'll want to be more detailed and sensitive about the condition, but if you're looking for a one word description:

volatile:
MW 1 a: characterized by or subject to rapid or unexpected change

Though correct, this might sound a bit too technical and cold in the context of the health of a human being.

His condition is on a knife-edge.

on a knife-edge [phrase]

To be on a knife-edge means to be in a situation in which nobody knows what is going to happen next, or in which one thing is just as likely to happen as another.

[Collins Cobuild]

There is at least a strong connotation of impending disaster:

on a knife-edge (also on a ˈrazor’s edge):

in a very dangerous or difficult situation where there is a risk of something terrible happening:

• He was balanced on a knife-edge between life and death.

[Farlex Partner Idioms Dictionary]

Others have given good answers—I'm particularly fond of "touch and go," but only if the recovery is a life-and-death matter—and another more poetic and general expression I could give similar to "can turn on a dime" would be "[prone to] vicissitude," e.g. "the vicissitudes of an itinerant lifestyle."

A less highbrow version of the same is "ups and downs."

The situation is 'hanging by a thread':

Be in a highly precarious state.

or

to be in a very dangerous situation or state : to be very close to death, failure, etc.
Example: The patient's life was hanging by a thread.

Dictionary.com gives a few more examples, and the origin, emphasizing that this idiom fits your prerequisite of the change being able to happen "without notice or reason":

Also, hang by a hair. Be in a risky or unstable situation, as in His promotion was hanging by a thread, or With the lead actor sick, the success of our play hung by a hair.
This expression, already proverbial in the early 1500s, alludes to Damocles, who vexed King Dionysius with constant flattery. The king invited him to a banquet where Damocles found himself seated under a naked sword suspended by a single hair, symbolizing his insecure position at the court.

A lot of the other answers are good if you want to imply the situation is more bad than good. Eg "touch and go", "hanging by a thread". If you want to illustrate uncertainty while not favoring either direction a common phrase is "it could go either way". It's good for situations where you might use the phrase "knives edge" but don't want to sound so dire or dramatic.

Lots of good answers already but I'll add:

The situation is fragile.

fragile

1. ADJECTIVE If you describe a situation as fragile, you mean that it is weak or uncertain, and unlikely to be able to resist strong pressure or attack.[journalism] ...moves that will place added strain on an already fragile economy.
   The Prime Minister's fragile government was on the brink of collapse. His overall condition remained fragile.
   — collinsdictionary.com

To say that things are "up in the air" seems to covey this meaning, a situation in which many factors are at work and it is impossible to predict how things will resolve themselves. At the same time, there is a sense that some sort of resolution is imminent.

I would not rely on one idiom for the message you are trying to convey. I'm not sure there is one that fully carries the idea, and it's an important message that you don't want to be misunderstood. I would just say something like what you said in the question:

Don't take anything for granted when you're taking care of [person]. They can seem fine one minute, and just a bit later their condition can be much worse. [And you can be more specific than just "condition can be much worse."]

The most common way to describe a situation that changes frequently and unpredictably is to say that "the situation is fluid".

I don't think this sort of thing is what you are really after, though, because the patient's condition is not really a situation. Sure, it creates a situation, but to warn about the instability of that situation instead of the instability of the patient's condition shows a self-centered disregard for the patient.

I think the most common way to warn your caretaker would be to say that the patient's "condition is unstable".

The idiom I would use to describe someone who is (perhaps) recovering but the situation could worsen at any time is

not out of the woods yet

"Despite improvement, not yet completely free from difficulties or danger. Often said in reference to someone's health or financial situation." [The Free Dictionary]

So you could counsel your friend that it's great that the patient has shown signs of recovery but that they are not out of the woods yet.

The generic most commonly used idiom to describe this situation is Anything could/can/might happen

Any outcome is possible The Free Dictionary

Anything's possible Cambridge Dictionary

The person's status is "in flux" or "fluctuating".

From Cambridge Dictionary:

flux (noun): continuous change:

Our plans are in a state of flux at the moment.