A aeolian (or A phrygian) in the verse and then changes to D dorian in the chorus...
Yes, the stable tones will change. At the very least, the most stable tone, the tonic, changes.
Keep in mind the difference in your question title and the description in the body. You really need to know both tonic and mode. You would then simply compare what tones are common and whether those tones change in terms of stability.
Stable tones are the tones of the tonic triad. If you rank them it would be tonic, dominant, mediant in decreasing stability. The related topic is tendency tones.
If the tonic doesn't change, but the mode does, then by definition only the modal degrees change and for the most part the stable tones are the same. The tonic and dominant don't change. The mediant would change.
If the change of tonic/mode is by perfect fifth (above or below) or to a relative major or minor key/mode, there will be some overlap in stable tones. Ex.
C major and
A minor, tones
E will be stable in both. Or from
C minor to
G is stable in both.
Nevertheless, even when there is overlap of stable tones, if the tonic changes, the functional role of each tone changes by necessity, because the functions/identities are relative to the tonic. The tone
G as a dominant changing to
G as a tonic don't function the same way. Theoretically their stability isn't exactly equal.