My understanding of the explanation in the text, given the example below it, is that from one position there is the potential for dissonance with respect to subsequent notes, i.e. in a "horizontal" manner, as opposed to the more typical "vertical" one. For instance, F#-F is clearly a dissonant interval. When it occurs "vertically" (or, harmonically) the dissonance is very obvious. But when it occurs "horizontally" (i.e. melodically) as in the manner shown above, it is still dissonant (i.e. unpleasant), but the effect is not as pronounced (in terms of unpleasantness), and therefore more subtle / harder to analyse, unless one is looking for "horizontal" dissonant patterns specifically.
The dissonance in the given example is even more subtle, since the dissonance does not occur along the same melodic line; in other words, neither the soprano nor the tenor melodic lines contain dissonant intervals by themselves in isolation, but when the two are put together harmonically, you have a dissonant interval formed by the F# from the soprano line 'leading' to an F in the tenor line (what the author refers to here as "cross-relations").
Contrary to harmonic dissonance, when dealing with melodic dissonance, clearly, until we've heard the next note in the series, we can only talk about the "potential" for dissonance, since we do not know if the resulting melodic interval is dissonant or not until the next note in the series has actually occured.
The implication in this passage is, however, that there are certain chords or harmonic intervals, that simply due to their position or configuration they have a higher potential for melodic dissonance, in the sense that there is presumably a higher number of subsequent patterns from that position that would lead to dissonance. Therefore it is not only important to find the right subsequent notes that do not lead to dissonance, but it is also important to pick chords / configurations that give you more flexibility in the first place, such that you have a larger variety of subsequent patterns to chose from later that will not be dissonant.
PS. Not strictly related to this, but you might find this video interesting, offering a more mathematical interpretation of what it means for harmonies or melodies to be "dissonant", and the physical basis of their unpleasantness. Or this one if you're feeling a bit more hardcore!