Let $(\mathcal{K},\otimes,\mathbf{1})$ be a locally small symmetric monoidal triangulated category, and let $D$ denotes the dual functor.
Now if $\mathcal{A}$ is a replete subcategory of $\mathcal{K}$, that is, $\mathcal{A}$ is isomorphism-closed. Is it possible to prove that $D(\mathcal{A})$ is still replete?
The origin of the question is the proof of proposition 2.6 in http://www.math.ucla.edu/~balmer/research/Pubfile/Filtration.pdf.
You didn't ask for the same conditions in your question body as in the title, but even in a rigid tensor triangulated category the result needn't hold. For instance in the derived category of finite dimensional vector spaces over $\mathbb{R}$ the unit ($\mathbb{R}$ in dimensions $0$) is not equal to the dual of any chain complex, since elements of each chain group in a dual complex are functions between real vector spaces to $\mathbb{R}$, but the elements of $\mathbb{R}$ are something else (e.g. sets of rational numbers.)
This is pretty pointless, but it does give a counterexample to your Proposition 2.6 as stated, taking $\mathcal{J}$ to be the whole category. One should probably define the operation $D$ to include repletion when applied to a thick tensor ideal.