It is very unlikely.
Neutrino detection is very inefficient. If a neutrino flux passes through a detector approximately 1 m deep, then only around one in a trillion would be detected - the exact number would depend on the energy of the neuitrinos and the type of material used for the detector, but it is a reasonable ball-park. So an animal the size of a blue whale that devoted ninety percent of its body to neutrino detection would likely 'see' no more than one solar neutrino per day - even if it had 'magically' evolved a detector that was somehow shielded from all the other sources of radiation in the environment. So imagine a very cold whale that lurks 10 km down in the deepest sea trench.
It wouldn't evolve in the first place. With such an astoundingly low detection rate, there wouldn't appear to be any possible evolutionary driver to induc ethe whale to develop a detector in the first place. Animal senses evolve to provide the animal with a benefit - seeing, hearing, tasting etc. are all useful for daily survival. 671
Detecting an occasional neutrino hardly appears to be of any use in any likely environment. In principle, if the environment had lots of localised nuclear reactions going on (like in a nuclear reactor), then the neutrino flux would be higher and it might almost be possible to detect it and use it as a warning to keep away - but if that was then case then it would be vastly more efficient to 'evolve' ionising radiation detectors (like little organic geiger detectors).
It would be almost impossible even if you tried to genetically engineer it. Real neutrino detectors, as well as being very large (100s of tonnes and upwards), require shielding from other radiation and the highest purity materials. There is sufficient natural radiation in any practical food-stuff that the animal's own body would produce false positive counts that would almost certainly outweigh any real neutrino detection rate.
But if you really want to detect neutrinos biologically... If you (or your test animal) was somehow transported to within a few million km of a sun just before it underwent a supernova, and you could organize a few thousand km of shielding to avoid damage from the massive outpouring of x-rays, gamma-rays, charged particles etc, then that shielding would hardly block the neutrino flux at all. And supernovas fling out such a massive neutrino flux that, even with the miniscule chance that any particular neutrino would react with your (or its) body, it would be sufficient to cause massive radioation damage. You wouldn't heat up noticeably, but you would die of radiation poisoning pretty quickly (depending on yuour e4xact distance from teh supernova). In such conditions, I believe it would be very possible for you to 'detect' the neutrinos via the Cherenkov radiation flashing within the vitreous humor in your eyballs. 435