For starfish as a whole, after they've run out of food (or at least out of nutritionally acceptable food), it generally takes them several months to starve to death (I'm talking like 6-8+ months, possibly upwards of a year for certain species), so many people think their starfish are doing fine, then, the next thing they know, their starfish are dropping legs and dying. - livinlifeinBKK, I suspect this will eventually happen to yours in the 10 gallon (if I'm wrong, in complete seriousness, please let me know after having it for around 18 months). The slow starvation is why people say they're hard to keep - they're hard to keep long term (i.e. for the duration of their natural life spans [some sand sifters have been reported to live up to 10 years old] or even for just a couple of years).
To the best of our knowledge, livinlifeinBKK is correct that Linckias, Fromias, etc. eat biofilms (some of them eat sponges and ascidians - i.e. tunicates, sea squirts, etc. - too), but that's why most people recommend a 75 gallon tank as a minimum for most of them - in the hopes that the tank will produce enough feed to keep them healthy. From everything I've read, it seems unlikely a ten gallon will produce enough long-term for the star (again, I hope I'm wrong, and please let me know if I am). Again, because they only starve after cleaning the tank out of food, some people even keep them for 2-3 years before they die of starvation. Not trying to be a downer here, it's just what I've found in my research so far.
With regards to Echinaster sentus specifically, from what I’ve been able to find, they eat bivalves (clams, mussels, scallops, etc.), sponges (they accept some species that have chemical defenses against fish, but it seems they prefer those with little to no chemical defenses period), possibly some anemones, and possibly other sessile (immobile) inverts too (so it may at some point go for corals, or it might not - there's not much information available on the diet of this species). You could probably supplement feed it using things like clams and sponges (carolina.com carries some relatively cheap sponge packages), but there's no guarantee it will survive long term. If you decide to try it, let us know how it works out after about 18 months - again, I'd be genuinely curious.