If you discount every single acute event, like infections and traumas (and for which there are good treatments nowadays), I would say that the diseases that would affect your immortals the most would be degenerative diseases, like dementia, amiloidosis and vascular diseases.
Even if the DNA doesn't suffer damage, the natural entropy of the continuous functioning of the organism will produce deterioration.
The bones will deteriorate so much that the immortal will eventually suffer from severe osteoarthrosis, rendering your immortal unable to move.
The neurons will also deteriorate eventually and lead to dementia.
Debris from the constant metabolism will eventually accumulate on the organs, inhibiting their normal functioning through mechanisms of amiloydosis
Also, even if the immortal has perfect sugar or lipid control, eventually the blood vessels will be coated with hidrophobic fat, leading to atherosclerotic events. Sugar or other molecular radicals will bond the natural molecules of the blood vessel wall and produce damage. Lastly, blood vessels will calcify or rupture from shear continuous usage.
The blood vessels and the heart will also undergo processes of remodeling to acomodate to continuous blood flow, loosening their elastic properties. These processes will lessen the efficacy of the heart, or produce aneurisms... which could lead to fatal hemorrhages or cardiovascular events if the immortal is allocated to a situation of physical or psychological stress.
Just like the blood vessels may be blocked by atherosclerosis, the bile will cristalize in the gallbladder and these crystals will block the biliary ducts, which could lead to fatal cholangitis and cholecystitis
Please note that these disorders will occur even in the absence of genetic damage, but the body will be unable to repair these lesions (at least if we're talking about a human body that is similar to our own, except for the DNA repair mechanisms). Old people without cancer die from these disorders. Machines, that by definition, don't have genes, cease to be able to operate after some years due to simple overusage.
In short, the body will exhaust itself from the continuous functioning in the long run and the immortal will suffer from every kind of degenerative disease.