So here are the main problems I've encountered:
Slime trails and munched leaves. Yes, slugs will eat Oca, but I find it to be quite rare. All of that oxalic acid in the leaves is a natural defence, and they seem to move on after a few leaves at most.
Unidentified Leaf Lurgy
I think this is some kind of rust fungus. It usually only appears on plants that are stressed by heat and lack of moisture, and the plant recovers given more favourable conditions. Those growing in light shade do not seem to suffer this problem to the same degree.
Not a pest, just frost-damage. The outer leaves have been killed by a light air frost, but the stems are undamaged.
See also here for more on frost damage to foliage, and here for tuber damage.
This stem rot occurs at ground level, usually just browning the stem, but occasionally withering it all the way through, causing the foliage to die. I've seen this every year to some extent, so it may be something that survives my composting process, or which is permanently present in the soil.
I've seen healthy and diseased stems right next to each other without it spreading, although on one occasion I've had a (weak) plant completely killed. It looks similar to potato blackleg.
Now and again I've seen blackfly on stems and leaves, but they have never stayed long, usually moving on to some nearby preferred venue, such as broad beans or peas, so presumably they don't like the taste of Oca.
Although they don't do too much direct harm, there is always the concern that they may carry viruses from plant to plant, so as a precaution I squash 'em on sight.
Rats & Birds
I've had an instance of rats and birds (I think ring-necked parakeets) scratching up and damaging tubers during very hard weather. I think this was only because of the desperate conditions, but it is worth watching out for swelling tubers pushing themselves up out of the ground where they could be an obvious target for hungry vermin.
So, quite a short list compared with diseases of, say potatoes. Though of course the list is probably not complete yet!