Probably, it depends on what happens to demand and production.
If U.S. demand falls enough so that U.S. demand is less than U.S. production, then U.S. producers will compete for that market and drive price down to production costs or world price whichever comes first. In that case, I wouldn't expect a price rise from American producers. A floor price would be established at the world price because U.S. producers could export to the world market instead of selling domestically.
If U.S. demand exceeds U.S. production, then U.S. producers will be free to charge whatever they want competition free up to the tariffed import price or to the point where U.S. demand = U.S. supply, whichever comes first. A ceiling price would be the tariffed world price.
I doubt I'd advocate tariffing oil for a couple of reasons. Energy is a big component of our economy, and it's going to have some dramatic impacts on our way of life. With unemployment at 25% I wouldn't want to tariff oil unless I thought it's put an awful lot of people to work. If oil is really a finite resource, it's better to keep our oil in the ground, as long as we don't liquidate the country to buy other's oil.
Would the same thing happen to any tariffed import?