It is true that a first-order sentence, which is preserved by finite direct products, is also preserved by infinite direct products; see Corollary 6.7 of S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57-103.

In H. J. Keisler's terminology, a first-order sentence is called a product sentence if it holds in a product $X\times Y$ whenever it holds in $X$ and $Y$, a factor sentence if it holds in $X$ and $Y$ whenever it holds in $X\times Y$. If memory serves, rather complicated characterizations of product sentences and factor sentences were found by Keisler in the 1960s; the work of Keisler's student J. M. Weinstein may also be relevant. I'm pretty sure the results or at least the references are in the book Model Theory by C. C. Chang and H. J. Keisler. (I happen to have at hand a reference to Weinstein's dissertation: Joseph M. Weinstein, First order properties preserved by direct product, University of Wisconsin, Madison, 1965.)

Now, if you wanted to know which first-order sentences are preserved by reduced products (direct products reduced modulo a filter on the index set, like ultraproducts but with any old filter instead of an ultrafilter), the answer (also due to Keisler) is very nice: a first-order sentence is preserved by proper reduced products if and only if it's logically equivalent to a Horn sentence. ("Proper" here means that the index set is nonempty and the filter is a filter of nonempty sets; if you want to include improper reduced products, insert "strict" before "Horn sentence".) I'm sure this is discussed in the Chang-Keisler book.

It is true that a first-order sentence, which is preserved by finite direct products, is also preserved by infinite direct products; see Corollary 6.7 of S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), 57-103.

In H. J. Keisler's terminology, a first-order sentence is called a product sentence if it holds in a product $X\times Y$ whenever it holds in $X$ and $Y$, a factor sentence if it holds in $X$ and $Y$ whenever it holds in $X\times Y$. If memory serves, rather complicated characterizations of product sentences and factor sentences were found by Keisler in the 1960s; the work of Keisler's student J. M. Weinstein may also be relevant. I'm pretty sure the results or at least the references are in the book Model Theory by C. C. Chang and H. J. Keisler. (I happen to have at hand a reference to Weinstein's dissertation: Joseph M. Weinstein, First order properties preserved by direct product, University of Wisconsin, Madison, 1965.)

Now, if you wanted to know which first-order sentences are preserved by reduced products (direct products reduced modulo a filter on the index set, like ultraproducts but with any old filter instead of an ultrafilter), the answer (also due to Keisler) is very nice: a first-order sentence is preserved by proper reduced products if and only if it's logically equivalent to a Horn sentence. ("Proper" here means that the index set is nonempty and the filter is a filter of nonempty sets; if you want to include the improper reduced product, insert "strict" before "Horn sentence".) I'm sure this is discussed in the Chang-Keisler book.