Suppose $X_n$ are finite sets for any natural integer $n$. let $Y$ be an infinite subset of $\prod_n X_n$. Do there exist $y$ and $y'$ in $Y$ and an infinite subset $S$ of $\mathbb N$ such that $y_n=y'_n$ for all $n$ in $S$?

What about $X_n=\left\lbrace 1,2,3,...,n\right\rbrace$, and $Y$ consisting of the sequences $\left(1,1,1,...,1,2,3,4,5,6,...\right)$ with $n$ ones for all $n\in\mathbb N$? The first true thing that comes into my mind when I hear "Pigeonhole Principle for infinite case" are some theorems in infinite Ramsey theory, such as this one. 


First, let me improve upon the countable counterexample of Darij Grinberg by giving an uncountable counterexample Y. Indeed, I shall give finite sets X_{n} and a subset Y of the product ΠX_{n} having size continuum (that is, as large as possible), such that any two distinct y, y' in Y have only finitely many common values. Let X_{n} have 2^{n} elements, consisting of the binary sequences of length n. Now, for each infinite binary sequence s, let y_{s} be sequence in the product ΠX_{n} whose n^{th} value is sn, the length n initial segment of s. Let Y consist of all these y_{s}. Since there are continuum many s, it follows that Y has size continuum. Note that if s and t are distinct binary sequences, then eventually the initial segments of s and t disagree. Thus, eventually, the values of y_{s} and y_{t} are different. Thus, y_{s} and y_{t} have only finitely many common values. So Y is very large counterexample, as desired. A similar argument works still if the X_{n} grow more slowly in size, as long as liminfX_{n} = infinity. One simply spreads the construction out a bit further, until the size of the X_{i} is large enough to accommodate the same idea. That is, if the liminf of the sizes of the X_{n}'s is infinite, then one can again make a counterexample set Y of size continuum. In contrast, in the remaining case, there are no infinite counterexamples. I claim that if infinitely many X_{n} have size at most k and Y is a subset of ΠX_{n} having k+1 many elements, then there are distinct y,y' in Y having infinitely many common values. To see this, suppose that Y has the property that distinct y, y' in Y have only finitely many common values. In this case, any two y, y' must eventually have different values. So if Y has k+1 many elements, then eventually for sufficiently large n, these k+1 many sequences in Y must be taking on different values in every X_{n}. But since unboundedly often there are only k possible values in X_{n}, this is impossible. In summary, the situation is as follows: Theorem. Suppose that X_{n} is finite and nonempty.
In particular, if the X_{n} become increasingly large in size, then there are very bad counterexamples to the question, and if the X_{n} are infinitely often bounded in size, then there is a very strong positive answer to the question. 


I think that what you are asking for is impossible. Given any element of $\prod_n X_n$ the element is uniquely determined by its image in each of the individual $X_n$'s. So if two elements of $Y$ agree on each $X_n$ then they must be the same element. In language similar to yours, what you probably want for the finite case is "If $X$ and $Y$ are finite sets such that $X < Y$ and $f:Y\rightarrow X$ is any map then there exists an element $x\in X$ such that $f^{1}(x) > x$." More generally, given any finite sequence $X_1,\ldots,X_n$ of finite sets and any set $Y$ such that $Y > X_1\cdotX_2\cdotsX_n$ and any sequence of maps $f_i:Y\rightarrow X_i$ then there exists a sequence of elements $x_1,\ldots,x_n$ and two elements $y,y'\in Y$ such that $f_i(y) = f_i(y')$ for any $i$. The problem with the infinite case is that there are injective but not surjective maps between infinite sets with the same cardinality. However, it is true that given a sequence of finite sets $X_1,X_2,\ldots$ and a set $Y$ with cardinality greater than that of $\prod X_n$, if you have any sequence of maps $f_i:Y\rightarrow X_i$ then there exists an uncountable subset $Z\subseteq Y$ such that for any two elements $z,z'$ of $Z$ you have $f_i(z) = f_i(z')$ for all $i$. In even more generality, I believe that if you have any set of sets $\{X_\alpha\}$ and any set $Y$ such that the cardinality of $Y$ is larger than the cardinality of $\prod_\alpha X_\alpha$ then you have a similar statement. 

