Apologies for being a bit late, but let me try to expand on my comment. An excellent source for this material is James Lewis' "user-friendly" survey article [Lew14].
First recall that for $X$ smooth over a field $k$,
\begin{equation*}
K_{n}(X)\otimes\mathbb{Q}\cong\bigoplus_{i}H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i))\,.
\end{equation*}
Now suppose $k=\mathbb{C}$. Then we have the (Betti) cycle class map
\begin{equation*}
\mathrm{cl}_{i,n}:H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i))\rightarrow H^{2i-n}(X,\mathbb{Q}(i))
\end{equation*}
whose image lands in the "Hodge" part of cohomology, i.e. in $\mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i)))$. Note that, as per my comment, this latter group is trivial if $X$ is also projective and $n\geq 1$.
Originally it was conjectured that $\mathrm{cl}_{i,n}$ is always surjective (the so-called Beilinson-Hodge conjecture, see [Bei86, Conjecture 6]). First note that the case $n=0$ of the Beilinson-Hodge conjecture is equivalent to the Hodge conjecture (clear in the projective case, some work for quasi-projective $X$). But Jannsen [Jan90, Corollary 9.11] found counterexamples in the case $n=1$ (and counterexamples exist for all $n\geq 1$). Indeed, the surjectivity of $\mathrm{cl}_{i,n}$ is tied up with the injectivity of certain higher Abel-Jacobi maps (see [Jan90, Corollary 9.10] and [dJL13, Theorem 4.9] more generally), and we know from Mumford that Abel-Jacobi (even $\otimes\mathbb{Q}$) can drastically fail to be injective in codimension $\geq 2$. Note, however, that the Bloch-Beilinson conjectures say that the higher Abel-Jacobi maps $\otimes\mathbb{Q}$ should be injective when $X$ is defined over a number field, so one could imagine that the Beilinson-Hodge conjecture is true for $X$ defined over number fields, and indeed if the Bloch-Beilinson conjectures and the Hodge conjecture are true then $\mathrm{cl}_{i,n}$ is indeed surjective when $X$ is defined over a number field (see [Sai09]).
For an example of $\mathrm{cl}_{i,n}$ failing to be surjective for $n\geq 1$, consider a K3 surface $X$ over $\mathbb{C}$ with points $P,Q$ such that $[P]\neq [Q]\in\mathrm{CH}^{2}_{\mathrm{deg}=0}(X)$. Let $U:=X\backslash\{P,Q\}$. Then $\mathrm{cl}_{2,1}:H^{3}_{\mathrm{Mot}}(U,\mathbb{Q}(2))\rightarrow H^{3}(U,\mathbb{Q}(2))$ is the zero map. Notice that $U$ is not defined over a number field here by the assumption, because Beilinson-Bloch predicts that $\mathrm{deg}:\mathrm{CH}^{2}(X)\xrightarrow{\simeq}\mathbb{Z}$ for $X$ defined over a number field (and it is known that $\mathrm{dim}_{\mathbb{Q}}\mathrm{CH}^{2}(X)_{\mathbb{Q}}=\infty$ for $X/\mathbb{C}$).
Note as well that along the Milnor diagonal $n=i$ there is evidence that the Beilinson-Hodge conjecture might be true even outside of the situation that $X$ is defined over a number field. See [AK09], [AS07], [Sai04] and [Sai09] for examples and further discussion.
In fact, $\mathrm{cl}_{i,n}$ factors through the cycle class map $\mathrm{cl}^{\mathcal{H}}_{i,n}$ into absolute Hodge cohomology:
\begin{equation*}
\require{AMScd} \begin{CD} H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i)) @>\mathrm{cl}^{\mathcal{H}}_{i,n}>> H^{2i-n}_{\mathcal{H}}(X,\mathbb{Q}(i))\\ @. {_{\rlap{\ \mathrm{cl}_{i,n}}}\style{display: inline-block; transform: rotate(30deg)}{{\xrightarrow[\rule{4em}{0em}]{}}}} @VVV\\ @. \mathrm{Hom}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n}(X,\mathbb{Q}(i))) \end{CD}
\end{equation*}
and the vertical arrow is a surjection (the kernel is the intermediate Jacobian $\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))$). So you are really asking about the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$. There the Beilinson-Hodge conjecture says that the image of $\mathrm{cl}_{i,n}^{\mathcal{H}}$ is dense, i.e. $\mathrm{cl}_{i,n}$ is surjective and the induced Abel-Jacobi map
\begin{equation*}
\mathrm{AJ}_{i,n}:H^{2i-n}_{\mathrm{Mot}}(X,\mathbb{Q}(i))_{\mathrm{hom}}\rightarrow\mathrm{Ext}^{1}_{\mathrm{MHS}}(\mathbb{Q}(0),H^{2i-n-1}(X,\mathbb{Q}(i)))
\end{equation*}
has dense image. For the reasons given above, if $n\neq 0,i$ then this conjecture is for $X$ defined over a number field.
Finally, one could speculate that the new ideas of Clausen-Scholze (see Clausen's modified Hodge Conjecture) have something to say in this setting too. ¯$\backslash$(ツ)/¯
Bibliography:
[AK09] D. Arapura, M. Kumar, Beilinson-Hodge cycles on semiabelian varieties, Math. Res. Lett. 16 (4) (2009), 557-562.
[AS07] M. Asakura, S. Saito, Beilinson's Hodge conjecture with coefficients for open complete intersections, in Algebraic Cycles and Motives 2, Edited by J. Nagel and C. Peters, LNS 344 (2007), 3-37, London Math. Soc.
[Bei86] A. A. Beilinson, Notes on absolute Hodge cohomology, Contemp. Math., 55, American Mathematical Society, Providence, RI, 1986, 35–68.
[dJL13] R. de Jeu, J. D. Lewis, Beilinson's Hodge conjecture for smooth varieties, J. K-Theory 11 (2013), no. 2, 243–282.
[Jan90] U. Jannsen, Mixed motives and algebraic $K$-theory, Lecture Notes in Math., 1400, Springer-Verlag, Berlin, 1990.
[Lew14] J. D. Lewis, Hodge type conjectures and the Bloch-Kato theorem, Contemp. Math., 608, American Mathematical Society, Providence, RI, 2014, 235–258.
[Sai09] M. Saito, Hodge-type conjecture for higher Chow groups, Pure and Applied Mathematics Quarterly 5 (3) (2009), 947-976.
[Sai04] S. Saito, Beilinson's Hodge and Tate conjectures, in Transcendental Aspects of Algebraic Cycles, Edited by S. Müller-Stach and C. Peters, LNS 313 (2004), 276-290, London Math. Soc.
