Is the Hodge Conjecture an $\mathbb{A}^1$-homotopy invariant?

Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutative). So it remains to note that all the ingredients of your question are "motivic". For this purpose one may recall that Chow motives embed into Voevodsky one and note that both the cycle classes and the Hodge classes are determined by Chow motives of the corresponding varieties.

This is more of a very long comment, but I believe so.

A modern description of the motivic homotopy category is $sPre(Sm/k)[N^{-1}][L^{-1}]$ where $N^{-1}$ is localizing with respect to the topology and $L^{-1}$ is localizing with respect to all the maps $X \times \mathbb{A}^1 \rightarrow X$. Chow groups of course satisfy Nisnevich descent, and are $\mathbb{A}^1$ invariant, and so they give a motivic sheaf. The question is then if the Hodge classes can be extended to a motivic sheaf. Since the definition of Hodge classes is only for smooth and projective $X$, it's not totally obvious what to use for a general smooth variety.

Here we should use the fact that Betti realization $X \mapsto X^{an}$ is a motivic sheaf of spaces, and that both the weight-filtered , I'll write $\hat{\mathcal{W}_k}A_{\log}$, and also $\mathcal{F}^p \cap \hat{\mathcal{W}_k}A_{\log}$ the weight-and-Hodge-filtered de Rham complexes are perfectly good motivic sheaves of cochain complexes (up to q.iso). Here I'm using the notation of Bunke and Tamme from "Regulators and cycle maps...", where they define them carefully. Bunke and Tamme prove Zariski descent for the latter two in Lemma 3.6, but their method actually shows etale descent. $\mathbb{A}^1$-invariance can be checked on the level of cohomology. We can just take the set-theoretic conjugate to get a sheaf $\overline{\mathcal{F}^p} \cap \hat{\mathcal{W}_k}A_{\log}$. To just work with sheaves of cochain complexes (up to q.iso), let's look at the singular cochains functor instead of the Betti realization functor. That is, let's look at $X \mapsto Sing_\mathbb{Q}^\bullet(X)$. We have a diagram

$\require{AMScd}$ \begin{CD} \mathcal{F}^p \cap \hat{\mathcal{W}_{2p}}A_{\log} @>>> A_{\log} @<<< \overline{\mathcal{F}^p} \cap \hat{\mathcal{W}_{2p}}A_{\log} \\ @. @AAA \\ @. Sing_\mathbb{Q}^\bullet(X) \end{CD}

Whose homotopy limit sheaf should output the $p$-Hodge classes for input a smooth projective $X$.

I was initially reluctant to contribute an answer because I'm not an expert on $\mathbb{A}^1$ homotopy theory. But let me flesh out my first comment a little, and you can decide whether this is close to what you wanted. Let $K_0(Var)$ be the Grothendieck ring of all complex algebraic varieties. Let $\mathbb{L}$ denote the class of the affine line. The quotient $K_0(Var)/(\mathbb{L}-1)$ might be thought of as the Grothendieck ring of naive $\mathbb{A}^1$-homotopy classes. This maps to $K_0(Var)[\mathbb{L}^{-1}]$ and the Hodge conjecture "factors through" the second group by http://arxiv.org/abs/math/0506210