Demonic possession occurs when an evil spirit hijacks the biological system of an organism, corrupting it and taking control of its functions. Although powerful, the possession eventually breaks down the body, degrading it as time goes on, due to its incompatibility with the ethereal form. The demon must exert more of its strength to keep the body together and remain in the mortal world, eventually destroying it and being sent back to its plane. There are, however, special individuals who can act as perfect "containers", able to withstand the pressure of having an evil spirit inside them. These are called demonhosts, and are the perfect blend of human and demon in which the spirit can use its powers to full extent without damaging its hijacked form.

Demonic possession acts similar to a virus, which functions by incorporating its own genetic material into the genome of the infected cells. This infection spreads to other cells, destroying the original cell in the process and eventually spreads throughout the body and brain. This form of self replication occurs very quickly and degrades neural tissue in order to complete its life cycle. This leads to the eventual breakdown of the human form, as it begins to show signs of heavy mutation (por ejemplo: horns, scaly skin, spikes, etc).

With a demonhost, the viral infection functions the same way, but cannot be discovered by tests. It leaves cells undamaged and cannot be detected as it moves from cell to cell. Being able to leave no trace of itself prevents it from being noticed or discovered by onlookers and scientists as it subtly takes over its host and consumes its soul, which is completely aware of it but unable to do anything about it.

How could a virus operate in this way?

This is known as an Asymptomatic carrier - someone who never shows signs of the infection.

This is distinct from an incubatory carrier (not yet showing signs of infection, but does later) or a convalescent carrier (no longer shows signs of infection, but has done so previously. Often incorrectly consider themselves to be "cured")

It's a retrovirus

A retrovirus is a type of RNA virus that inserts a copy of its genome
into the DNA of a host cell that it invades, thus changing the genome
of that cell. Such viruses are either single stranded RNA (e.g. HIV)
or double stranded DNA (e.g. Hepatitis B virus) viruses.

Once inside the host cell's cytoplasm, the virus uses its own reverse
transcriptase enzyme to produce DNA from its RNA genome, the reverse
of the usual pattern, thus retro (backwards). The new DNA is then
incorporated into the host cell genome by an integrase enzyme, at
which point the retroviral DNA is referred to as a provirus. The host
cell then treats the viral DNA as part of its own genome, transcribing
and translating the viral genes along with the cell's own genes,
producing the proteins required to assemble new copies of the virus.
It is difficult to detect the virus until it has infected the host. At
that point, the infection will persist indefinitely.

In most viruses, DNA is transcribed into RNA, and then RNA is
translated into protein. However, retroviruses function differently,
as their RNA is reverse-transcribed into DNA, which is integrated into
the host cell's genome (when it becomes a provirus), and then
undergoes the usual transcription and translational processes to
express the genes carried by the virus. (ref)

A retrovirus is difficult to detect even with our current knowledge, until it has fully infected the host. Then it's next to impossible to get rid of.

While the enzyme retroviruses use, reverse transcriptase, was discovered in 1971, the first retrovirus wasn't discovered until 1979. Others, like HIV, weren't isolated until 1983.

Because retroviruses are all somewhat different from each other and can have long incubation periods, there are likely some we haven't figured out how to detect yet. There are some methods (see A simple, general method for detecting retroviral RNAs expressed in cells) to "search for unidentified retroviruses expressed in human cancer cells and tissues" but we'd have to know to look for them. Retroviruses are common and just finding one in a human doesn't mean it's demonic possession; it could just be a new strain of hepatitis.

If your demonhost isn't showing symptoms, there is no reason to test her/him. Some retroviruses lie dorment for years. If there are symptoms but they are explained by other medical (or psychological) conditions, there would be no advanced testing. Also, your society may be similar to ours in medical technology, but that doesn't mean identical. Perhaps they haven't discovered retroviruses or engineered tests for them yet.

In the case of the demonhost, it sounds like what you're describing is more of a symbiotic (rather than parasitic) relationship between the infection and the host. Unlike the possessed individual, in the case of the demonhost the viral infection in this case would just be using the host cells as as food source while it reproduces itself throughout the body, but doesn't go into the 'active mode' of possession.

Now, that having been said, if there's foreign genetic material in your demonhosts, then that foreign genetic material MUST BE detectable. You can't make DNA invisible. It's physically there, if you put it under a microscope, you can see there's something in there that's not normal human cells.

Now, that said, that doesn't necessarily cause a problem for your story. Just because it's POSSIBLE to detect it doesn't mean that physicians in your world know how to do it, or that it's easy or practical.

The simplest solution I can think of is that the demonic cells break down very rapidly when removed from the human body. Perhaps they're relying on body heat, the body's natural electromagnetic field, or both. In any case, it's very plausible to have a scenario where you have a VERY short window to examine a sample before the viral cells break down into bits of protein indistinguishable from the normal metabolic waste constantly in your bloodstream.

In this case the medical profession in your world would probably BELIEVE that the infection was undetectable even through they could detect it if they knew how.