This is a tricky question to answer, the main problem being that "inflammatory diseases" is one of those buzz-words that includes a vast and diverse range of illnesses. For any type of claim at this regard to be at least meaningful very precise distinctions have to be made and extreme care has to be made before doing any generalization (i.e.: if a compound helps relieving symptoms of arthritis, this does not imply it will help for inflammatory bowel syndrome).
In summary: there seems to be a possible acute effect of ω3 PUFA on certain inflammatory situations. Evidence of prevention of inflammatory diseases is scarce at least. In certain cases such as prostate cancer certain PUFA may however be a risk factor.
Further studies are warranted before being able to give a precise answer on this.
Here are few examples of studies to support this conclusion
This article gives a meta analysis of clinical trials on the role of ω3 in joint pain relieve.
A meta-analysis of the analgesic effects of omega-3 polyunsaturated fatty acid supplementation for inflammatory joint pain. - Goldberg and Katz - Pain, 2007
They conclude that there may be an effect in decreasing certain symptoms.
We conducted a meta-analysis of 17 randomized, controlled trials assessing the pain relieving effects of omega-3 PUFAs in patients with rheumatoid arthritis or joint pain secondary to inflammatory bowel disease and dysmenorrhea
Supplementation with omega-3 PUFAs for 3-4 months reduces patient reported joint pain intensity (SMD: -0.26; 95% CI: -0.49 to -0.03, p=0.03), minutes of morning stiffness (SMD: -0.43; 95% CI: -0.72 to -0.15, p=0.003), number of painful and/or tender joints (SMD: -0.29; 95% CI: -0.48 to -0.10, p=0.003), and NSAID consumption (SMD: -0.40; 95% CI: -0.72 to -0.08, p=0.01). Significant effects were not detected for physician assessed pain (SMD: -0.14; 95% CI: -0.49 to 0.22, p=0.45) or Ritchie articular index (SMD: 0.15; 95% CI: -0.19 to 0.49, p=0.40) at 3-4 months. The results suggest that omega-3 PUFAs are an attractive adjunctive treatment for joint pain associated with rheumatoid arthritis, inflammatory bowel disease, and dysmenorrhea.
On the other hand this recent study found no effect of ω3 in the amelioration of symptoms peripheral arterial disease.
Effect of Omega-3 fatty acid supplementation on markers of platelet and endothelial function in patients with peripheral arterial disease. - Mackay et al. - Atherosclerosis., 2012
OBJECTIVE: Omega-3 fatty acids have been shown to reduce platelet and endothelial activation in patients with or at risk of cardiac disease. We aimed to determine if Omega-3 fatty acid supplementation in addition to best medical therapy can reduce the increased platelet and endothelial activity that is present in patients with intermittent claudication.
CONCLUSION: Supplementation with Omega-3 fatty acids had no affect on platelet and endothelial activation or markers of inflammation in patients with peripheral arterial disease.
Another study showed an amelioration in the inflammatory status of cancer patient with fish oil
Fish oil supplement alters markers of inflammatory and nutritional status in colorectal cancer patients. - Silva et al. - Nutr Cancer., 2012
This study aims at investigating whether there are changes in inflammation markers and/or in nutritional status of patients with colorectal cancer undergoing chemotherapy who were supplemented with fish oil.
Low doses of fish oil supplement can positively modulate the nutritional status and the C-reative protein/albumin ratio.
So, are these studies confirming the claim that ω3 "may help lower risk of chronic diseases such as heart disease, cancer, and arthritis."?
Definitely not. What these and other similar studies show is that they can lower inflammations markers during an already existing inflammatory state. They do not say anything about preventing or lowering the risk of inflammation, let alone of cancer or arthritis.
At this regard, I could find this study where the effect of ω3 on the risk of prostate cancer were monitored over the course of 7 years.
Serum phospholipid fatty acids and prostate cancer risk: results from the prostate cancer prevention trial. - Brasky et al. - Am J Epidemiol., 2011
Inflammation may be involved in prostate cancer development and progression. This study examined the associations between inflammation-related phospholipid fatty acids and the 7-year-period prevalence of prostate cancer in a nested case-control analysis of participants, aged 55-84 years, in the Prostate Cancer Prevention Trial during 1994-2003. Cases (n = 1,658) were frequency matched to controls (n = 1,803) on age, treatment, and prostate cancer family history. Phospholipid fatty acids were extracted from serum, and concentrations of ω-3, ω-6, and trans-fatty acids (TFAs) were expressed as proportions of the total.
No fatty acids were associated with low-grade prostate cancer risk. Docosahexaenoic acid was positively associated with high-grade disease
The study findings are contrary to those expected from the pro- and antiinflammatory effects of these fatty acids and suggest a greater complexity of effects of these nutrients with regard to prostate cancer risk.
Similar conclusions are drawn by this review:
Dietary N-6 and N-3 polyunsaturated fatty acids and prostate cancer risk: a review of epidemiological and experimental evidence. - Astorg - Cancer Causes Control., 2004
RESULTS: There is no or little evidence of an association of linoleic or arachidonic acids with PCa risk. Most epidemiological studies failed to find an association of PCa risk with fish or long-chain n -3 PUFA intake, but two recent cohort studies did find an inverse association of fish consumption with the risk of the latest stages of PCa. alpha -linolenic acid intake was associated with an increase of PCa risk in a majority of epidemiological studies, but other studies did not find this association. Experimental work in vitro and in vivo, as well as mechanistic studies, support a protective effect of long-chain n -3 PUFA on PCa, but data on the effects of linoleic and alpha -linolenic acids are scarce.
CONCLUSIONS: Long-chain n -3 PUFA from fish are possible promising nutrients for the dietary prevention of PCa, but to-date with little epidemiological support. In contrast, studies suggest that alpha -linolenic acid intake might be a risk factor. New work, both epidemiological and experimental, is awaited to clarify these results.