Technically yes, the FAA should not permit US airlines to operate to countries with which it has no diplomatic relations. I expect that AA will suspend flights and that the FAA will deny Venezuelan airlines the right to fly to the US - although this may be delayed due to the shutdown and in order to not strand people in either nation.
In principle, I agree, however, there are a couple of notable exceptions:
1) When U.S. diplomatic relations with China were switched from Taiwan to the People's Republic, commercial flights between Taiwan and the United States continued. The relationship between the two sides is complicated, but in this "family dispute", as it was described to me by my college professor on Chinese History, this is no reason to disrupt the flow of commerce, and the mutual benefit to maintaining a dialogue far outweigh any short-term military solution. Yes, there's the "KLM-Asia" stuff, and the PRC government demanding things from airlines serving China such as referring to Taiwan as part of China, but in general, there are many carriers at TPE with whom the airlines' home countries have no diplomatic relations.
2) From January, 1961, to July, 2015, the United States had no diplomatic relations with Cuba, and yet many airlines flew charters there. A special situation, definitely, but one I always found interesting that the charter rights weren't given to a third party, such as Mexico. Mehh, politics - I hope I never understand that logic.
I hope something can bring Venezuela back into stability. If things stabilize, I'll bet carriers will be tripping over themselves to return.