Predicates of infinite arity
I have several thoughts about this question.
First, to my way of thinking, there is little difference between an infinite-ary relation $R(a_0,a_1,\dots)$ on a set $X$ and a unary relation on a suitable power of that set, such as $X^\omega$ or $X^\alpha$. For example, an $\omega$-ary relation on $\{0,1\}$ is essentially the same thing as a unary relation on Cantor space $2^\omega$. In the one case, we have $R(a_0,a_1,\dots,a_n,\dots)$, and in the second case we have $R(\langle a_0,a_1,\dots,a_n,\dots\rangle)$. It is a mere stylistic difference without substantive difference. An $\omega$-ary relation on the natural numbers $\omega$ is essentially the same as a unary relation on Baire space $\omega^\omega$. A binary relation on Baire space is the same as an $(\omega+\omega)$-ary relation on $\omega$.
For these reasons, it seems to me that mathematics is filled with abundant examples of infinite-ary relations. The lexical $<$ relation on the Cantor set is essentially the same as an $(\omega+\omega)$-ary relation on the two-element set $\{0,1\}$. Binary relations on Baire space $\omega^\omega$ are essentially the same as $(\omega+\omega)$-relations on the natural numbers $\omega$.
I believe that we prefer in these cases to think of the infinite-ary relation as a unary or finite-ary relation on the higher-order space of sequences, for several reasons. First, it is easier to think of the relation as a unary relation in the higher order space of sequences, simply because we don't mind so much going to a higher-order and we are used to finite-arity relations. Secondly, the move to the higher order space allows us to be more precise about exactly which sequences are allowed to be considered. If one has an infinite-arity relation, but doesn't specify the extent of the second-order sequences that are to be considered (from which model of set theory will they be drawn?), then the ontological meaning of that relation is a little ambiguous. But when we think of the relation as a finite-ary relation on a certain space of sequences, specified by a set of sequences, then the extensional nature is more clear.
To give an example, usually one views the axiom of determinacy as concerned with games on $\omega$, so that the players construct a play of the game $a_0,a_1,a_2,\dots$, and the winning condition of a game is a unary condition on Baire space $R(\langle a_0,a_1,\dots\rangle)$. But one could just as easily view the winning condition as an $\omega$-ary relation on $\omega$, as $R(a_0,a_1,\dots)$. And this wouldn't really make any difference; it is an inessential stylistic syntactic difference.
But lastly, let me also point out that there is a literature on infinite-ary functions, undertaken for example by Addison, as in his theory of infinitary Boolean operations. I once had the pleasure of taking a seminar on the topic that he offered in Berkeley on the topic, and he considered many different Boolean operations $f:\{0,1\}^\omega\to\{0,1\}$, and investigated their nature.
Not sure if this is a comment or an answer:
In the study of infinitary language $L_{\infty,\kappa}$ we usually assume that predicate and function symbols have finite arities. I think there are two reasons for this:
1) Many theorems for $L_{\infty,\kappa}$ do not hold true if we allow infinite arities. For instance, the downward Lowenheim-Skolem theorem. The closure of a set of size $\kappa$ under a $\omega$-ary function maybe of size $\kappa^\omega$.
Recall that the Lowenheim-Skolem number of a logic is a fixed cardinal $\lambda$ such that any subset $A$ of a structure $M$ with $|A|=\kappa$ will be contained in a substructure of $M$ of size at most $\kappa+\lambda$. If $\lambda<\kappa$, then any $A$ must be contained in a substructure of size $|A|$. Clearly, this can not always be the case if $\kappa^\omega>\kappa$, which also hints that the set-theory starts playing a role.
2) A formula $\phi(\vec{x})$ where $\vec{x}$ is infinite will necessitate the use of infinitely long sequences of quantifiers, $\forall \vec{x} \phi(\vec{x})$, $\exists\vec{x} \phi(\vec{x})$, or $\forall x_1\exists x_2\ldots \phi(x_1,x_2,\ldots)$. This brings us to the study of the infinitely-deep languages $M_{\infty,\kappa}$, which historically followed the study of the languages $L_{\infty,\kappa}$, trying to remedy some of the restrictions.
See for instance: Maaret Karttunen, Model theoretic results for infinitely deep languages, Proceedings of the Finnish-Polish-Soviet logic conference (Polanica Zdrój, 1981) Studia Logica 42 (1983), no. 2-3, 223--241 (1984).
As far as I know, infinitely-deep languages do not exclude the usage of predicate and function symbols with infinite arity, but I wouldn't be surprised if "nicer" results hold true under the finite arity restriction.
I suppose this is relevant. Let $I$ be a set and $\mathcal{U}$ an ultrafilter over $I$. If $X$ is any compact Hausdorff space then any function $x: I \to X$ converges along $\mathcal{U}$ to exactly one point. This allows us to introduce infinitary operations $f_{I,\mathcal{U}}: X^I \to X$ defined by $$f_{I,\mathcal{U}}(x) = \lim_\mathcal{U} x.$$ It's kind of nice because a subset is closed under these operations iff it is topologically closed, a map from $X$ to $Y$ is an algebraic homomorphism iff it is continuous, and algebraic products equal topological products. In fact the class of compact Hausdorff spaces is a variety in the sense of universal algebra. The "free algebra" construction yields the Stone-Cech compactification, etc. This is a little paper I wrote on the subject when I was a graduate student. It doesn't really seem to go anywhere but I thought it was cute.