The relation between t-structures and derived category

Assume $\mathcal D$ is a presentable stable $\infty$-category with a $\mathrm t$-structure (which is accessible and compatible with filtered colimits), and let $\mathcal A$ be its heart, $\mathcal{D(A)}$ its derived $\infty$-category.

Note that under those hypotheses, $\mathcal A$ is Grothendieck abelian (Higher Algebra, 1.3.5.23.).

You get a natural inclusion functor $\mathcal A\to \mathcal D$, which you can extend to $Fun(\Delta^{op},\mathcal A)\to \mathcal D$ (by geometric realization).

This functor preserves weak equivalences : indeed, (see HA, 1.2.4.4. and 1.2.4.5.), if $X$ is a simplicial object of $\mathcal A$ (and therefore $\mathcal D_{\geq 0}$), there is a spectral sequence with $E^2_{p,q}=\pi_p\pi_q(X)$ converging to $\pi_{p+q}(|X|)$ in $\mathcal A$.

Since $\pi_q(X) = 0$ for $q\neq 0$ in our situation (as $X$ takes values in $\mathcal A$), this spectral sequence degenerates, and $\pi_p(|X|) = \pi_p(X_\bullet)$ (where the latter are homotopy groups as computed in $\mathcal A$ via the classical Dold-Kan correspondance)

It follows that this functor yields a (unique up to a contractible space of choices) functor $\mathcal D_{\geq 0}(\mathcal A)\to \mathcal D$ (via, again the Dold-Kan correspondance).

Now $\mathcal D$ is presentable and stable, and this functor $\mathcal D_{\geq 0}(\mathcal A)\to \mathcal D$ preserves colimits, so it extends (again, uniquely) to a functor $\mathcal{D(A) \to D}$ which preserves colimits and preserves $\mathcal A$ (HA 1.3.5.21. : $\mathcal{D(A)}$ is right complete with its classical $\mathrm t$-structure, which implies in particular that $\mathcal{D(A)} = \lim(\dots \overset{\Omega}\to \mathcal D_{\geq 0}(\mathcal A)\overset{\Omega}\to \mathcal D_{\geq 0}(\mathcal A))$, and then we use 1.4.4.5. which says that this precisely has the universal property of "presentable stabilization")

Now these hypotheses on $\mathcal D$ may look pretty strong but they're reasonable ; and in fact you can't really hope for much better : if $\mathcal D$ isn't presentable, then it could be something like $\mathcal D^{-}(\mathcal A)$ and then there's no hope to get a sensible functor $\mathcal{D(A)}\to \mathcal D^{-}(\mathcal A)$ (this is something that won't change whether you're in an $\infty$-categorical setting or not).

This is probably already somewhere in HA but I couldn't find it written down completely.

As I pointed out in the comments, under different hypotheses (which may look weaker), you can get away with a natural functor $\mathcal D^{-}(\mathcal A)\to \mathcal D$ (morally, this is because $\mathcal D^{-}(\mathcal A) \subset \mathcal{D(A)}$ is $\bigcup_n \mathcal D_{\geq n}(\mathcal A)$, and so it is determined by $\mathcal D_{\geq 0}(\mathcal A)$ via finite limits, so you don't need a presentability hypothesis- you do, however need some hypothesis on $\mathcal D$ to be able to replace the first step, since you can't take colimits as easily).

I guess there may also be a more general statement about $\mathcal D^b(\mathcal A)$, the bounded derived $\infty$-category, which should require less hypotheses, since "everything is finite" but I couldn't tell you on the top of my head.

I had reason to think about this a few years ago. When $\mathcal D$ arises as the derived category of an abelian category (with a possibly exotic $t$-structure), a construction of a realization functor $D^b(A) \to \mathcal D$ can be found already in Beilinson-Bernstein-Deligne-Gabber. (For them $\mathcal D$ is the derived category of constructible sheaves, equipped with the perverse $t$-structure, so $A$ is the category of perverse sheaves.) Their construction of the realization functor can in fact be imitated in the $\infty$-categorical setting, too, and in this case it works more generally for an arbitrary stable $\infty$-category $\mathcal D$. The way B-B-D-G construct the functor is they use the filtered derived category $\mathcal{D}F$. If $\mathcal D$ is the derived category of an abelian category $B$, then $\mathcal D F$ is the category of complexes in $B$ with a bounded filtration, localized at filtered quasi-isomorphisms. There is an induced $t$-structure on $\mathcal D F$ from a $t$-structure on $\mathcal D$, such that if the $t$-structure on $\mathcal D$ has heart $A$ then the heart of the $t$-structure on $\mathcal D F$ is isomorphic to the abelian category $\mathrm{Ch}^b(A)$ of bounded chain complexes in $A$. Thus we can consider the composition $\mathrm{Ch}^b(A) \to \mathcal D F \to \mathcal D$ where the first is the inclusion of the heart and the second forgets the filtration. A spectral sequence argument shows that this composition takes quasi-isomorphisms in $\mathrm{Ch}^b(A)$ to equivalences in $\mathcal D$, so there is an induced functor $D^b(A) \to \mathcal D$ which is the one we want.

The point of the above is that for a general triangulated category $\mathcal T$ there is no sensible triangulated category $\mathcal T F$ of filtered objects in $\mathcal T$, but if $\mathcal T$ happens to be the derived category of an abelian category we can write down the filtered derived category by hand. This is not a problem in the world of stable $\infty$-categories.

I believe that an analogous argument works also for complexes that are not necessarily bounded (using instead filtrations that are unbounded to the left or right or both). But if we want there to exist a functor $\mathcal DF \to \mathcal D$ that forgets the filtration then we need $\mathcal D$ to have sequential limits or colimits, and one should be careful with the spectral sequence argument.