The answer is yes if the $x_i$ are integers. Write
$$
B(X):=\prod_{i=1}^n (X-x_i),
$$
so that $B(X)$ is a monic polynomial in $\mathbf{Z}[X]$. For any $P(X)\in\mathbf{Z}[X]$, we can write
$$
P(X) = B(X) Q(X) + R(X)
$$
where $Q,R\in\mathbf{Z}[X]$ and $\deg(R)<n$. Here $R(x_i)=P(x_i)=y_i$ for every $i$. Since there is a unique polynomial $f(X)\in\mathbf{Q}[X]$ of degree less than $n$ which satisfies $f(x_i)=y_i$ for $i=1,2,\dots,n$, and both $R(X)$ and the Lagrange interpolation polynomial have the properties required of $f(X)$, it follows that the Lagrange interpolation polynomial equals $R(X)$ and hence has integer coefficients.

The answer is no if the $x_i$ are not integers. For instance put $x_1=y_1=1/2$ (and $n=1$), then $P(X):=X$ satisfies $P(x_1)=y_1$, but the Lagrange interpolation polynomial is $1/2$.