Yes, codeine and other opiate drugs can suppress tissue oxygenation. This occurs because codeine and other opiates suppress a wide variety of brain activities, including those that monitor and respond to changing gas levels in the blood. This process, known as respiratory depression, means that bodily mechanisms that normally would kick in when oxygen levels fall and carbon dioxide levels rise do not happen. Breathing remains shallow, heart rate remains slow, and the person does not attempt to change position. Most opiate overdose deaths are attributed to respiratory depression.
A 2006 study showed that individuals who have obstructive sleep apnea are at higher risk of respiratory depression from opiates than the general population. It is believed that the apnea has desensitised these individuals to low oxygen levels, so their internal mechanisms are more sluggish than normal; the addition of the opiate drug puts them at high risk of hypoxia.