The European Food Safety Authority investigated the health impact of casomorphins in their report "Review of the potential health impact of β-casomorphins and related peptides".
β-casomorphins can act on opioid receptors, but their effect is much weaker than that of your own opioids or opioid drugs:
Animal data clearly indicate that BCMs, including BCM7, can act as
opioid receptor agonists, probably acting via μ-type opioid
receptors. However, in most if not all animal studies to date, in
vivo opioid effects for BCM7 and related milk-derived peptides have
only been observed following intra-peritoneal (i.p.) or
intra-cerebro-ventricular (i.c.v.) administration. In comparison to
medicinal and endogenous opioids, bovine BCM7 does not seem to be a
very potent opioid ligand.
For the BCMs to act on our opioid receptors they first have to get there. Peptides are digested in the intestine, and they have to cross the intestinal epithelial barrier and the blood brain barrier to get to the opioid receptors. Those are significant barriers, and we don't know if BCMs can cross them in significant amounts:
Relatively little is known on the mechanisms of transfer of intact
peptides longer than 3 amino acids across the intestinal barrier. If
this transport occurs, then the extent is very low and passive
diffusion is the most likely transfer mechanism
We also don't know how much BCMs are actually in cheese. They are generated by proteolysis of casein, but they can also be degraded themselves by further proteolysis. There have been some studies trying to find out how much BCMs are in cheese with unclear or contradictory results:
Therefore, to date no data concerning the actual amounts of BCM7 in
cheeses can be obtained from the literature
As a conclusion the report states:
Based on the present review of available scientific literature, a
cause-effect relationship between the oral intake of BCM7 or related
peptides and aetiology or course of any suggested non-communicable
diseases cannot be established
As for an addictive effect of casomorphins, at least rats don't get addicted to it. In the article "An Assessment of the Addiction Potential of the Opioid Associated with Milk" the authors investigated the possible addictive effect of β-casomorphin on rats:
Consequently, systemically administered β-casomorphin has very limited
or no reinforcing properties similar to those of morphine. Ingestion
of milk products containing β-casomorphin is not likely to become the
focus of an addiction.
This study was the only result for "casomorphin addiction" in Pubmed. There does not seem to be any data on possible effects in humans.
From the limited data that exists we can't conclude that casomorphins are actually addictive in practise.