Note that in algebra one usually writes $H^{-n}$ instead of $H_n,$ so connective and coconnective algebras live in the same bigger category of (cohomologically graded) DGA's. (There is also a topological convention of writing $\pi_{-n}$ instead of $H^n$). If $A$ is connective is has cohomology in negative degrees, and there is a map $A\to H^0(A).$ If $A$ is coconnective, then there is a map $H^0(A)\to A.$

Now $K$ theory is properly an invariant of a(n $\infty$-) category, not an algebra. It is never interesting to take the category of all $A$-modules (whose $K$ theory is trivial by the Eilenberg swindle), so you always want to impose some finiteness condition. Usually the $K$ theory of $A$ is defined as the $K$ theory of the category of perfect modules. In this case, given a map $f:A\to B$ of DGA's, the functor of restriction $B\mathrm{mod}\to A\mathrm{mod}$ does not (in general) preserve perfect objects, so the only natural map on $K$ theory is the one associated to the induction functor, $M\mapsto M\otimes_A B$ from $A$-modules to $B$-modules. Induction from $H^0(A)\to A$ in the coconnective case does not induce an isomorphism on $K$ theory in the very simplest nontrivial coconnective dga, $k\oplus k[1]$ (square zero extention), where the induced map on $K^0$ is trivial (any perfect module over $k$ has equal total dimension of even and odd cohomology). In the connective case, I can't think of an obvious counterexample but I actually am not sure that what you have heard is true in general (though I think it is true if $A$ is assumed connected instead of connective).

However there are other categories of modules that one can study. In particular, in the coconnective case it is interesting to study the category of modules which are perfect over $H^0(A)$ (or, more generally, whose cohomology groups are finitely generated over $H^0(A)$); this is related to the coherent $K$ theory of a scheme.

Now in these cases, the restriction functor along $H^0(A)\to A$ gives a valid map on $K$ theory, and moreover induces an isomorphism on $K^0$ as well as on all higher $K$ groups.

Note that in algebra one usually writes $H^{-n}$ instead of $H_n,$ so connective and coconnective algebras live in the same bigger category of (cohomologically graded) DGA's. (There is also a topological convention of writing $\pi_{-n}$ instead of $H^n$). If $A$ is connective is has cohomology in negative degrees, and there is a map $A\to H^0(A).$ If $A$ is coconnective, then there is a map $H^0(A)\to A.$

Now $K$ theory is properly an invariant of a(n $\infty$-) category, not an algebra. It is never interesting to take the category of all $A$-modules (whose $K$ theory is trivial by the Eilenberg swindle), so you always want to impose some finiteness condition. Usually the $K$ theory of $A$ is defined as the $K$ theory of the category of perfect modules. In this case, given a map $f:A\to B$ of DGA's, the functor of restriction $B\mathrm{mod}\to A\mathrm{mod}$ does not (in general) preserve perfect objects, so the only natural map on $K$ theory is the one associated to the induction functor, $M\mapsto M\otimes_A B$ from $A$-modules to $B$-modules. Induction from $H^0(A)\to A$ in the coconnective case does not induce an isomorphism on $K$ theory in the very simplest nontrivial coconnective dga, $k\oplus k[1]$ (square zero extention), where the induced map on $K^0$ is trivial (any perfect module over $k$ has equal total dimension of even and odd cohomology). In the connective case, the map on $K$ theory goes from $K(A)\to K(H^0(A))$. I can't think of an obvious counterexample but I actually don't think think this functor induces an isomorphism on $K^0$ in general despite what you heard; I may be wrong.

However there are other categories of modules that one can study. In particular, in the coconnective case it is interesting to study the category of modules which are perfect over $H^0(A)$ (or, more generally, whose cohomology groups are finitely generated over $H^0(A)$); this is related to the coherent $K$ theory of a scheme.

Now in these cases, the restriction functor along $H^0(A)\to A$ gives a valid map on $K$ theory, and moreover induces an isomorphism on $K^0$ as well as on all higher $K$ groups.

Note that in algebra one usually writes $H^{-n}$ instead of $H_n,$ so connective and coconnective algebras live in the same bigger category of (cohomologically graded) DGA's. (There is also a topological convention of writing $\pi_{-n}$ instead of $H^n$). If $A$ is connective is has cohomology in negative degrees, and there is a map $A\to H^0(A).$ If $A$ is coconnective, then there is a map $H^0(A)\to A.$

Now $K$ theory is properly an invariant of a(n $\infty$-) category, not an algebra. It is never interesting to take the category of all $A$-modules (whose $K$ theory is trivial by the Eilenberg swindle), so you always want to impose some finiteness condition. Usually the $K$ theory of $A$ is defined as the $K$ theory of the category of perfect modules. In this case, given a map $f:A\to B$ of DGA's, the functor of restriction $B\mathrm{mod}\to A\mathrm{mod}$ does not (in general) preserve perfect objects, so the only natural map on $K$ theory is the one associated to the induction functor, $M\mapsto M\otimes_A B$ from $A$-modules to $B$-modules. Induction from $H^0(A)\to A$ in the coconnective case does not induce an isomorphism on $K$ theory in the very simplest nontrivial coconnective dga, $k\oplus k[1]$ (square zero extention), where the induced map on $K^0$ is trivial (any perfect module over $k$ has equal total dimension of even and odd cohomology). In the connective case, you can also cook up a counterexample (though this is more tricky). The paper linked in Maxime Ramzi's answer gives a very restrictive condition on coconnective $A$ which guarantees that the map is indeed an equivalence on $K$ theory.

However there are other categories of modules that one can study. If $A$ has finite-dimensional total cohomology, it is interesting to study the $K$ theory of the category of all $A$-modules with finite-dimensional total cohomology (which is bigger than the category of perfect objects); more generally one can also look at the related category of modules whose cohomology is finitely generated as an $H^0(A)$ module (a generalization of "coherent $K$ theory" of an affine scheme). In both of these cases, every $A$-module has a filtration (the cohomological filtration) whose associated graded subquotients are concentrated in a single cohomological degree. This means that pullback along the map $H^0(A)\to A$ is a valid functor which induces an isomorphism on $K^0$ (and, at least in the coconnective case, on higher $K$ groups as well).

Note that in algebra one usually writes $H^{-n}$ instead of $H_n,$ so connective and coconnective algebras live in the same bigger category of (cohomologically graded) DGA's. (There is also a topological convention of writing $\pi_{-n}$ instead of $H^n$). If $A$ is connective is has cohomology in negative degrees, and there is a map $A\to H^0(A).$ If $A$ is coconnective, then there is a map $H^0(A)\to A.$

Now $K$ theory is properly an invariant of a(n $\infty$-) category, not an algebra. It is never interesting to take the category of all $A$-modules (whose $K$ theory is trivial by the Eilenberg swindle), so you always want to impose some finiteness condition. Usually the $K$ theory of $A$ is defined as the $K$ theory of the category of perfect modules. In this case, given a map $f:A\to B$ of DGA's, the functor of restriction $B\mathrm{mod}\to A\mathrm{mod}$ does not (in general) preserve perfect objects, so the only natural map on $K$ theory is the one associated to the induction functor, $M\mapsto M\otimes_A B$ from $A$-modules to $B$-modules. Induction from $H^0(A)\to A$ in the coconnective case does not induce an isomorphism on $K$ theory in the very simplest nontrivial coconnective dga, $k\oplus k[1]$ (square zero extention), where the induced map on $K^0$ is trivial (any perfect module over $k$ has equal total dimension of even and odd cohomology). In the connective case, I can't think of an obvious counterexample but I actually am not sure that what you have heard is true in general (though I think it is true if $A$ is assumed connected instead of connective).

However there are other categories of modules that one can study. In particular, in the coconnective case it is interesting to study the category of modules which are perfect over $H^0(A)$ (or, more generally, whose cohomology groups are finitely generated over $H^0(A)$); this is related to the coherent $K$ theory of a scheme.

Now in these cases, the restriction functor along $H^0(A)\to A$ gives a valid map on $K$ theory, and moreover induces an isomorphism on $K^0$ as well as on all higher $K$ groups.