The answer is almost everyone can learn to sing.
There is a small percentage of people who suffer from amusia, which encompasses "tone deafness" and the ability to distinguish between different pitches. For such people, learning to sing is exceedingly difficult, because they cannot hear that they are making a mistake, so they can't even really tell what note they should sing, unless they "know." For such people, about the only thing I could imagine working would be for them to learn via "muscle memory"—how certain pitches "feel." Again, though, this is at best a limited technique, since muscle memory takes a while to develop, and also might not work for distinguishing very close pitches (I suppose this depends on the individual in question).
Putting such individuals in a choir is also a very ineffective way of teaching them to sing, because of the amount of time that might be needed to correct their mistakes (which they can't hear). The most extreme example of this I have encountered was in a university choir I sang in a few years ago. An alto somehow managed to sing in an "audition octet" with some of the biggest voices in the choir—thereby masking the fact that she wasn't actually able to sing on pitch. (She was singing very quietly, which apparently didn't help finding this problem.)
However, it was very obvious that there was something wrong—there would frequently be "wrong notes" sticking out in chords during warm-ups, and often during the main rehearsals. It was easy to hear from nearby, but got lost in the sounds of a 120-person choir singing loudly. If the repertoire had not included a performance of Mahler's Second Symphony (Auferstehung), with its cruelly exposed a cappella passages, it probably would have gone undiscovered. In that particular work, though, an alto randomly singing a minor third or so above the rest of the section stuck out rather painfully. The singer in question was quietly asked not to participate in the performance.