None of the terms 'dissonance', 'consonance', 'stable' or 'unstable' have precise and universally agreed-upon definitions in a musical context, so when people make statements like this, there's always the question of what they really mean. To me, consonance/dissonance is more about a momentary sensation of smoothness/roughness, while stability and instability are more a question of the listener's expectations - do they hear the harmony as 'wanting to go somewhere'?
a chord can be stable, even when dissonant
An obvious example of that might be the tonic seventh chord in blues.
I'm curious as to whether the opposite can also be true.
Perhaps something as simple as a major V triad in major tonality? It's consonant in and of itself, but with its leading note, very much wants to 'go somewhere' (to most listeners, at least).
I immediately think of a chord built with notes G, C, and E. This, to me, sounds consonant, as it is an inverted C major chord, yet it also wants to "resolve" to that same chord with a different inversion. However, common practice period ideology would say that the perfect fourth within the original chord is dissonant.
I think that's partly because that definition of 'dissonant' includes something of the notion of being 'unstable', although inversions of chords do also actually sound less consonant too, as the set of harmonics is further away from matching the harmonic series of the lowest note.