No, the "leaves" in the DNA section (these are Shared Ancestor Hints) never go away, unless you or the other person changes or removes something in your respective trees (or removes them altogether).
https://support.ancestry.com/s/article/DNA-Shared-Hints
The leaves in your tree, not in the DNA section, are record/tree hints, nothing to do with DNA. They won't go away unless you click "no/ignore" and even then, they will remain in that individual's "ignored hints" section. Or, if you accept the hint and add it to your tree, the hint goes away but the record is now attached to your tree.
https://support.ancestry.com/s/artic...091979344-3686