Note: The question originally asked about infinite direct sums.
If you meant "infinite direct product", on the other hand, then the answer is "no." Consider $\mathbb{Z}$-modules; then a module is projective if and only if it is free. So $\mathbb{Z}$ itself is projective, but $\prod_{i=1}^{\infty}\mathbb{Z}$ is not free, hence not a projective $\mathbb{Z}$-module.
Here's the answer to the original question:
The direct sum of free modules is free, even if the family is infinite.
Assume that $\{P_i\}_{i\in I}$ is a family of projective modules; for each $i$, there exists $M_i$ such that $P_i\oplus M_i$ is free. Then $\left(\bigoplus_{i\in I}P_i\right) \oplus \left(\bigoplus_{i\in I}M_i\right) \cong \bigoplus_{i\in I}(P_i\oplus M_i)$ is a direct sum of free modules, hence free. Since there is a module $M$ (namely, $M=\oplus M_i$) such that $(\oplus P_i)\oplus M$ is free, it follows that $\oplus P_i$ is projective.
That is, any direct sum of projective modules is projective.