A Congressional "No" vote would not be legally binding.

[cthulu2016]

Congress would have to pass a law suspending the war powers act relevant to Syria, expressly say no to strikes, and outlaw use of funds for such strikes.

And even then it wouldn't necessarily be binding, though it would be a strong effort in that direction. It would give the Supreme Court something serious to chew on, at least.

Congress would be free to consider violation of that law a "high crime" and impeach the President, but their interpretation of "high crime" is unlimited, so that could equally apply to spitting on the sidewalk.

Long story short: Constitutional scholar Rand Paul is mistaken. Congress merely failing to pass this resolution means nothing except politically.