Denying passengers due to body odour is in the category of things that happen, but happen rarely enough to sometimes be considered newsworthy.

Here's a fairly straightforward example from 2010:

Smelly Passenger Kicked Off Plane

Man's body odor on an Air Canada Jazz flight overwhelmed his fellow passengers.

...The Air Canada Jazz flight from Charlottetown, on Prince Edward Island, was preparing for the two-hour flight to Montreal when passengers on the plane reportedly complained about the odor coming from one of their fellow travelers. The crew eventually decided to ask the man to leave the plane...

...[airline spokesperson] confirmed that a passenger was removed from the Feb. 6 afternoon flight but would not say exactly why the passenger was asked to leave, saying the airline "must respect the privacy of our passengers.

It seems to be a subjective judgement call to be made by the crew, which of course leaves it open to abuse or accusations of abuse, for example (2020):

Family kicked off flight over alleged body odor sues American Airlines

A Michigan couple kicked off an American Airlines flight last year because of their alleged body odor filed a federal lawsuit against the carrier claiming they were victims of religious and racial discrimination...

..."Mr. Adler asked the agent if the body odor was emanating from him, his wife or child and the agent would not respond to the question but continued to state that the Adlers must leave the plane at the instruction of the pilot and because they had extremely offensive body odor," the suit says.

The Adlers told airline staff they had bathed early that day, prior to their morning flight.

"In response, defendant's agent made disparaging and derogatory statement telling the Adlers that he knew Orthodox Jews take baths once a week," Nwogu wrote.

As for appearing visibly sick, airlines and national agencies have their own policies regarding assessing "Fitness to Fly" (e.g. here's the UK's guidance) and may require passengers with certain conditions to have an assessment made and/or bring a medical certificate.

The International Air Transport Association (IATA) have a "medical manual" with general guidance on matters including "Efficiently deal with sick passengers", but this is a reference resource rather than standardized policy.

Ultimately, it comes down to a case-by-case judgement call by the crew. It's not uncommon for flights to be diverted or forced to make an emergency landing due to a passenger appearing critically ill or in need of treatment.

Here's a tragic and grisly example from 2018 of both events:

Passenger taken off plane for 'horror' smell dies

The passenger whose illness caused a Transavia flight to make an emergency landing last month has died.

[The passenger] was diagnosed with tissue necrosis — which causes the premature death of cells in living tissue.

...In a statement to Euronews earlier this month, the airliner Transavia confirmed the emergency landing was because of "medical reasons," but did not say if it was due to the man's odour.

However, due to privacy legislation, the airline said it "cannot disclose details about individual cases."

Captains can decline to carry anyone they believe may pose a risk to flight safety. Extremely overweight passengers are declined, I believe, on the grounds that the chairs and seatbelts cannot ensure their safety, and thus of those around them.

Sick passengers could easily be declined if there was a risk they might infect others, or that their condition might deteriorate so as to require a diversion. Someone with "extreme body odour" might be declined if their scent would be unendurable by those around them (if this could ever happen), but short of that it's hard to imagine someone's smell being a threat to safety.

Conditions are not specifically laid down, so it would very much depend on the circumstances.