The World Health Organization indicates the COVID-19 genome is detected in assays. Once a patient tests positive, is there any reason not to expect that future instances of the same assay will result positive?
There are two general categories of tests typically used for identifying viral infections: those based on viral genetic material and those based on host immune response.
The CDC page on MERS-CoV testing (also a coronavirus, just not the one that causes COVID-19) is an informative resource to start with (bold added and reformatted by me):
Molecular tests are used to diagnose active infection (presence of MERS-CoV) in people who are thought to be infected with MERS-CoV based on their clinical symptoms and having links to places where MERS has been reported.
Real-time reverse-transcription polymerase chain reaction (rRT-PCR) assays are molecular tests that can be used to detect viral RNA in clinical samples.
Serology testing is used to detect previous infection (antibodies to MERS-CoV) in people who may have been exposed to the virus. Antibodies are proteins produced by the body’s immune system to attack and kill viruses, bacteria, and other microbes during infection. The presence of antibodies to MERS-CoV indicates that a person had been previously infected with the virus and developed an immune response.
For some diseases, the primary type of test used is measuring immune reactivity; tuberculosis comes to mind.
However, tests currently being used for 2019-nCoV are of the "molecular" type: RT-PCR assays detecting viral RNA. Viral RNA is only present during an active infection. These tests can be used both when a patient is showing symptoms, to identify the infection, and also to retest later to show that the infection has been cleared.