It's true, but... well. There is more to it. We are not actually doing that yet, as far as I know. US Gov has sent officials to Japan to discuss the possibility of having their spare refinery capacity used. That was May, it's unclear if we are doing that now. I have found nothing on if that is occurring.
That said, oil exports happen all the time. Oil is sold on a global market. A commodities market, where commodities traders, buy and sell oil on a global market.
There is no such thing as a country that only exports, or only imports oil, unless it's under government control. Because if it isn't, then a private company might be able to get a cheaper bid, even if they are in Saudi Arabia, buying from Egypt. Everything is bought and sold on the Commodities market. If a company in Canada places a higher bid for a shipment of oil, then that US company will ship it north. That is what a global commodity is about.
That said, there is an issue with refineries. We don't drill for enough oil, yes. So we must import oil. But, even despite our stagnate refining capacity, other countries are much worse off. Mexico, with a socialized government run oil industry, can't fund the building of refineries. We buy oil from them, and they buy refined diesel and other oil products from us routinely. Canada, also having a eco-nut contingent, also lacks refining capacity, and thus they buy refined oil products from us too.
My personal favorite, Brazil being cited as an Ethanol success story, is one of the largest importers of diesel. It wasn't even 3 years ago, they were claiming Brazil was eliminating energy imports, now they are one of the top 5 importers of US refined oil.
Lastly, and this struck me as insane, but not all of our refineries... are ours. Venezuela actually owns 3 refineries sitting along the gulf of mexico. Guess where that gas goes?
Point being, we are hitting a shortage of refining capacity. It's not all for us though. We're selling fuel everywhere, and because of that demand, it's driving us to shortages here. We do need more refineries.