The most important problem that could occur would be the lack of a single voice in creating foreign policy. Imagine a situation like the one we have today. We have a Senate that is controlled by the Democrats while the Republicans control the House. If Congress were in control of foreign policy, it would be extremely difficult to have any sort of a coherent policy. We know that it is hard for Congress to do anything in domestic policy, so we can assume they would not agree on foreign policy issues either.
A foreign policy controlled by Congress, then, would be uncoordinated with many different voices clamoring to be heard. It would also be tardy as endless debates in Congress would make it hard for any definitive action to be taken quickly.