I never suffocated myself so not entirely sure, but when you suffocate, it's painful, right? But of course different kind of pain than being injured or sick. What I'm wondering is, if the "painful" (or suffering) experience of suffocation involves the standard pain pathway mediated by nociceptors? Or is it some other kind of pain pathway due to a lack of oxygen? I don't mean psychological pain.
This is more of a hypothesis
I am not sure if suffocation per se would cause pain. Asphyxia, as RoryM indicated in their comments, can lead to anxiety and panic but not really pain. However, forceful breathing may lead to muscular fatigue which may result in pain. Pain induced by muscle fatigue is called myalgia. Myalgia is possibly triggered by low pH generated by lactic acid, via ASIC3 (Acid Sensing Ion Channel) receptor. Carbon dioxide has no role in this process either directly or as a synergist.
Alan R. Light, Charles J. Vierck, and Kathleen C. Light. (2010) Translation from Mouse Sensory Neurons to Fibromyalgia and Chronic Fatigue Syndromes. Boca Raton, FL: CRC Press; Chapter 11 (Myalgia and Fatigue)
Being no expert on pain, I will share some thoughts on the issue.
According to the following site(http://www.helpforpain.com/arch2000dec.htm), there are two types of pain: nociceptive and neuropathic pain. Neuropathic pain involves the central and peripheral nervous system, a possibility I would discount due to no apparent link to suffocation.
Thus, it is likely that, if "traditional pain" is felt, nociceptors are involved. A possible pathway is the decrease in oxygen, possibly detected by certain nociceptors which respond to lead to pain. However, I have not suffocated either, not met anyone suffocating so this is only a possibility.