Background SARS‑CoV‑2 (COVID-19) causes olfactory dysfunction which is characterized by anosmia or hyposmia. Characterization of olfactory dysfunction has added value to the diagnosis and prognosis of the disease. Nevertheless, scarce information exists about COVID-19 patients suffering from olfactory dysfunction in Iraq. This study aimed to identify olfactory dysfunction (anosmia or hyposmia) in Iraqi COVID-19 patients and examine their response to smell exercise at Baghdad Medical City Complex, Baghdad, Iraq. Methodology This case series prospective study involving 300 patients (160 males and 140 females) with COVID-19 infection was conducted from June 1, 2020, to October 1, 2021. We recorded signs and symptoms of COVID-19 among patients by examining olfactory dysfunction, n-butanol olfaction test, and smell test exercise. Results Anosmia and hyposmia were found in 69.3% and 30.7% of the patients, respectively; of these, 65.7% were of sudden onset. The association between olfactory dysfunction and smoking was not significant. The most frequent signs and symptoms of COVID-19 were fatigue, fever, loss of taste, myalgia, headache, sore throat, cough, depressed appetite, dyspnea, nausea, abdominal pain, and diarrhea. The highest frequencies of occurrence of anosmia (30.7%) and hyposmia (13.3%) were in the age group of 31-40 years. The majority (47.7%) of patients with olfactory dysfunction recovered within one month of COVID-19 onset. The rest of the patients recovered within one month to 16 months. The most commonly encountered ear, nose, and throat symptoms were nasal obstruction, rhinorrhea, and facial/ear pain. The percentages of patients with anosmia and hyposmia recovering with smell exercise were significant at 64.7% and 25.3%, respectively. Conclusions The prognosis of olfactory dysfunction in COVID-19 patients was good as most cases recovered within a short period with concomitant smell exercise. Olfactory dysfunction in the majority of COVID-19 patients was self-limiting in young age groups, albeit in association with the non-severity of the disease. Being an important public health issue, examining olfactory dysfunction aspects should be considered in the diagnosis, prognosis, and treatment protocols of COVID-19 patients. In-depth exploration is needed to examine olfactory and gustatory dysfunction in patients suffering from severe COVID-19.