Here is a tentative definition based on finitary unbiased monoidal categories – I make no claims of usefulness!
An infinitary unbiased monoidal category consists of the following data:
An ordinary category $\mathcal{C}$.
For each small ordinal $\alpha$, a functor $T_\alpha : \mathcal{C}^\alpha \to \mathcal{C}$, which maps a $\alpha$-sequence $(A_0, A_1, \ldots)$ to $A_0 \otimes A_1 \otimes \cdots$.
A natural isomorphism $\textrm{id}_\mathcal{C} \Rightarrow T_1$.
For each partition $\alpha = \sum_{i < \gamma} \beta_i$, a natural isomorphism
$$T_\gamma \circ (T_{\beta_0} \times T_{\beta_1} \times \cdots ) \Rightarrow T_\alpha$$
such that for each double partition $\alpha = \sum_{i < \gamma} \sum_{j < \delta_i} \beta_{i,j}$, where $\beta_i = \sum_{j < \delta_i} \beta_{i,j}$ and $\delta = \sum_{i < \gamma} \delta_i$, the two composites
\begin{multline}
T_\gamma \circ (T_{\delta_0} \circ (T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times T_{\delta_1} \circ (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \newline
\Rightarrow T_\gamma \circ (T_{\beta_0} \times T_{\beta_1} \times \cdots) \Rightarrow T_\alpha
\end{multline}
and
\begin{multline}
T_\gamma \circ (T_{\delta_0} \circ (T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times T_{\delta_1} \circ (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \newline
\Rightarrow T_\delta \circ ((T_{\beta_{0,0}} \times T_{\beta_{0,1}} \times \cdots) \times (T_{\beta_{1,0}} \times T_{\beta_{1,1}} \times \cdots) \times \cdots) \Rightarrow T_\alpha
\end{multline}
are equal, and the two composites
$$T_\alpha \Rightarrow T_\alpha \circ (T_1 \times T_1 \times \cdots ) \Rightarrow T_\alpha$$
$$T_\alpha \Rightarrow T_1 \circ T_\alpha \Rightarrow T_\alpha$$
are the identity natural transformation on $T_\alpha$.
An infinitary lax monoidal category is what we get if we replace "natural isomorphism" by "natural transformation" in the above. An infinitary strict monoidal category is what we get if we replace "natural isomorphism" by "identity".
This is just a straightforward generalisation of the definition appearing in [Leinster, Higher operads, higher categories, §3.1]. The reason why this even makes sense is that ordinal addition is associative – if it weren't, we'd be stuck. This leads us to our first easy example:
Example. The category of small ordinals (and monotone maps) is a infinitary strict monoidal category under ordinal addition.
Less tautologically:
Example. Any (co)complete category is an infinitary unbiased monoidal category under (co)products.
(Though, in some sense we assumed this in our definition...)
Conjecture. Martin's third example probably works if we assume that $X \otimes (-)$ preserves sequential colimits, and that the chosen morphisms $I \to X$ are sufficiently nice. First, by the Mac Lane–Kelly coherence theorem, we can make a finitary biased monoidal category into an unbiased one, and we can use the colimit construction to define the infinitary monoidal products. We use the assumption on $X \otimes (-)$ to prove that, e.g.
$$A_0 \otimes (A_1 \otimes A_2 \otimes \cdots) \cong A_0 \otimes A_1 \otimes A_2 \otimes \cdots$$
The universal property of colimits should be enough to ensure the coherence of the infinitary associators – but I haven't checked. We need to assume something about the choice of morphisms $I \to X$ so that
$$I \otimes I \otimes I \otimes \cdots \cong I$$
holds. (For example, choosing a non-isomorphism for $I \to I$ is a bad idea!)
Non-example. Take $\mathcal{C}$ to be the category of small ordinals, and take $T_\alpha$ to be ordinal addition for all finite ordinals $\alpha$, and $T_\alpha = 0$ for all infinite ordinals $\alpha$. This is not an infinitary unbiased monoidal category, because
$$T_\omega \circ (T_1 \times T_1 \times T_1 \times T_0 \times T_0 \times \cdots ) \ncong T_3$$
and so we see that there is some form of continuity required.
Also, a reminder about a famous trick:
Remark. Suppose we define infinitary monoids to be discrete infinitary strict monoidal categories. There are non-trivial small infinitary monoids: for example, take any complete small semilattice with $\sup$ as the monoid operation. However, any infinitary monoid that is also a group must be the trivial monoid: after all, for any element $x$,
$$x^{-1} \cdot (x \cdot x \cdot x \cdot \cdots) = (x^{-1} \cdot x) \cdot (x \cdot x \cdot x \cdot \cdots) = x \cdot x \cdot x \cdot \cdots$$
and then cancel $x \cdot x \cdot x \cdot \cdots$ on both sides of the equation to obtain $x^{-1} = \textrm{id}$.
(EDIT) Non-example. The tensor product of modules doesn't give an infinitary monoidal product. To be precise, if we define
$$\textrm{Hom}_R (A_0 \otimes_R A_1 \otimes_R A_2 \otimes_R \cdots, B) \cong \textrm{Multi}_R(A_0, A_1, A_2, \ldots ; B)$$
where by $R$-multilinear we mean that $f(\ldots, r a, \ldots) = r f (\ldots, a, \ldots)$, then I don't see how $R \otimes_R R \otimes_R R \otimes_R \cdots$ can be isomorphic to $R$. (A multilinear map $R \times R \times R \times \cdots \to B$ is not necessarily determined by just what it does to $(1, 1, 1, \ldots)$, unlike the finite case.)
Now, some closing remarks:
How do we define an "infinitary braided/symmetric monoidal category" as extra structure on top of an infinitary unbiased monoidal category? The trouble is that a permutation of an infinite ordinal can change its order type (e.g. $\omega + \omega_1 = \omega_1 \ne \omega_1 + \omega$)... but this probably isn't a big problem. It seems to me that the morally correct way of defining an "infinitary unbiased symmetric monoidal category" would define a functor $S_\kappa : \mathcal{C}^\kappa / \textrm{Sym}(\kappa) \to \mathcal{C}$ for each cardinal $\kappa$ and some natural isomorphisms to the various $T_\alpha$.
Is there a sensible notion of an "infinitary biased monoidal category"? We would need to define $T_\alpha$ for at least $\alpha = 0$, $\alpha = 2$, and every infinite regular ordinal $\alpha$. This is still a large amount of data! For the finitary fragment we can adopt the same coherence axioms, but I don't know what the coherence axioms for the infinitary fragment ought to be. There will have to be some new ones that don't show up in the finitary fragment: For example, we would need an axiom like
$$I \otimes I \otimes I \otimes \cdots \cong I$$
for the monoidal unit $I$ – because inductively applying the left unitor can never delete infinitely many copies of $I$; or we could instead make
$$A_0 \otimes A_1 \otimes \cdots \otimes A_n \otimes I \otimes I \otimes I \otimes \cdots \cong A_0 \otimes A_1 \otimes \cdots \otimes A_n$$
into an axiom, because we can't even apply the right unitor here!
