What exactly would happen to the body if the immune system did not detect or respond to a the common cold virus like Rhinovirus?
This was answered very well by @anongoodnurse in Biology.SE question What are the effects of the common cold in an immunodeficient person? plus there was a study by Bowden (1997) which indicated that:
rhinovirus was responsible for 25% of community-acquired VRIs [viral respiratory infections] among bone marrow transplant recipients.
In one small study by Greenberg (2003), rhinovirus in the immunocompromised led to significant mortality from lower respiratory infection:
Among high-risk patients with cancer, rhinovirus infections are often fatal. In a study of 22 immunocompromised blood and marrow transplant recipients who were hospitalized with rhinovirus infections, 7 (32%) developed fatal pneumonia. The remaining patients had infections confined to the upper respiratory tract. In 6 of the 7 fatal cases, rhinovirus had been isolated in bronchoalveolar lavage fluid or an endotracheal aspirate before death.
Note that this is in hospitalised patients; it says nothing of non-hospitalised patients.
This conclusion has been disputed according to @anongoodnurse, but it can be seen to corroborate Greenburg's 2003 study.
In a slightly larger study by Murali et al. (2009) among people with hematological cancers:
Respiratory viral pathogens are a common cause of morbidity in patients with hematologic malignancies. ...Both a rapid viral culture with direct fluorescence antibody (DFA) staining and a PCR-based assay (MultiCode-PLx Respiratory Virus Panel) were performed on patients with hematologic malignancies, who underwent collection of a nasopharyngeal swab or bronchoalveolar lavage from October 2006 to April 2007. Eighty-two samples from 70 patients were obtained; all patients had upper respiratory tract symptoms. Respiratory viruses were detected in 10 samples (12%) by conventional virological methods and in 31 samples (38%) by the MultiCode-PLx assay. ...40% of these patients had pneumonia in addition to the upper respiratory tract symptoms. [emphasis added]
@anongoodnurse noted that there is no mention of mortality, but that does not mean a proportion didn't die from pneumonia, just like Greenberg (2003) may have found that those with pneumonia had upper respiratory tract infections. He may have just not mentioned it in light of the fact that he wanted to highlight the mortality rate of pneumonia in his cases.
@anongoodnurse summarised by saying
So long story short, they have stuffy, runny noses, sore throat, cough, etc. Clearly the virus itself causes damage to the mucosa; that is integral to viral replication. After entering a mucosal cell, the virus replicates, then the progeny virus is released by lysis of the cell. This damage itself causes inflammation (not the same as an immune reaction), pain, etc. The major difference seems to be a more severe and prolonged experience.
for which I would add that fatal pneumonia can develop from the prolonged infection of rhinovirus.
Rhinovirus infections, although usually limited to the upper respiratory tract, can extend beyond the oropharynx and may cause complications in the lower respiratory tract, including pneumonia (Imakita, et al. 2000).
Bowden, R. A. (1997). Respiratory virus infections after marrow transplant: the Fred Hutchinson Cancer Research Center experience. The American journal of medicine, 102(3), 27-30. doi: 10.1016/S0002-9343(97)00007-7 pmid: 10868139
Greenberg, S. B. (2003). Respiratory consequences of rhinovirus infection. Archives of internal medicine, 163(3), 278-284. doi: 10.1001/archinte.163.3.278
Imakita, M., Shiraki, K., Yutani, C., & Ishibashi-Ueda, H. (2000). Pneumonia caused by rhinovirus. Clinical infectious diseases, 30(3), 611-612. doi: 10.1086/313723
Murali, S., Langston, A. A., Nolte, F. S., Banks, G., Martin, R., & Caliendo, A. M. (2009). Detection of respiratory viruses with a multiplex polymerase chain reaction assay (MultiCode-PLx Respiratory Virus Panel) in patients with hematologic malignancies. Leukemia & lymphoma, 50(4), 619-624. doi: 10.1080/10428190902777665 pmid: 19373660