Iridology claims that it's practitioners can diagnose various illnesses via looking at the eyes of patients. Is there evidence that supports or debunked their claims?
No, according to an evaluation in 1979:
A systemic review undertaken in 1999 also concluded iridology does not work:
However apart from scientific studies a skeptic may be satisfied that iridology does not work by a simple observation: if the practice was as effective as claimed, naturopaths would use iridology as the initial step in diagnosing and treating patients without requesting further information. It would be a superb way to differentiate their services from mainstream medicine and would quickly impress patients.
However this does not happen. Natural health practioners, like doctors, generally require new patients to complete a survey outlining their medical history, diet, and present complaints. The introduction of iridology into a natural health session may be more useful as a secondary aid in medical theatre, creating the sense that the practioner has additional powers of observation that ultimately lend more weight to the lifestyle instructions they deliver to patients.
As for iridology specifically...
There are many conditions which can be diagnosed by looking at the eye, and for the most part, they are only conditions which affect the eye. Pupillary response can be measured to assess brain function and jaundice can be detected by looking into the eye. However, that is not the claim here. It is the sclera (white part) which becomes jaundiced, not the iris, and it is the pupil not the iris which becomes the indicator of brain function.
There is also an argument which can be made, stating that since the PAX6 gene codes for the formation of the iris and also the frontal lobe, there could be a theoretical link between the iris and one's personality, but only very, very theoretically. There is absolutely no proposed link between disease in one organ and an expression in the already genetically coded external appearance of the iris. Nor is there an idea of any way in which there could be such a link.
Aside from that, I don't believe that there are many more peer-reviewed, legitimate published studies on-topic. I would suggest you look into genetics, anatomy and physiology for an understanding of why the claims of iridologists do not even begin to meet the burden of proof.
Regarding pseudoscienctific claims in general....
Such claims are the reason boards like this need to exist and the skeptical community needs to keep at it.
In medical and pharmaceutical research time, talent, and funding are all very limited. Simple ethics dictate that these limited resources are are allocated to the areas with the most potential for benefit. Put simply: designing, implementing and testing ideas with a valid base in science in order to do the most good.
Most professionals realize that it is unethical to waste limited resources on claims which don't even merit consideration because they are simply ludicrous and based on imaginary principles.
In short, if it can be disproven with only a basic knowledge of human anatomy and physiology, or doesn't have a reason to be taken seriously in the first place, it is usually (and rightly) dismissed out of hand and without comment by the professionals.
As a result, it can sometimes be hard for the public to find published studies which EXPLICITLY DISPROVE pseudoscientific ideas. Sadly, people who sell these products and push these ideas exploit the knowledge and ethics of the scientific community, knowing that real scientists will not waste the time and resources to refute the claims they make. So, while it may sometimes be hard to find an accurate layman's debunking of a specific claim, this in NO WAY lgitimizes it, because the burden of proof lies on the one making the claim.