The point of the example in the question in the title is that, to my knowledge, there are no minimal pairs that contrast [ð] and [θ] in English, yet, if someone pronounced a word with those sounds correctly we would think it sounds wrong or at least think they must be using a different dialect than us. It is not free variation, and it is not phonologically conditioned.
I believe 'x' and 'ng' also behave this way in English, although in these cases we are dealing with two segments, always with 'x', and sometimes with 'ng'. Actually it is not about the orthography, so I should probably talk about [ks]/[gz] and [ŋ]/[ŋg]. You can correct me if I am wrong, but I don't know of any words that contrast only on these sounds. In this case there my be some phonological motivation for [ks] vs [gz]. I haven't thought it through. I know in the case of 'ng' there is some morphological motivation, but I think that would still qualify for the phenomena I am asking about.
I am studying a language which has prenasalized many words that begin with voiced consonants, but some words do not get prenasalized. I don't believe there are any minimal pairs to be found where one the only difference is prenasalization at the beginning of the word.