Presidents, republican or democrat, have very little, if any, authority or influence over the price of gas or the economy. So they can take the credit or blame, but can't really affect either.
Where presidents do have considerable impact is as the leader of their party. From the "bully pulpit" they can carry the mail for the party's agenda (typically social or international). In recent memory, think Clinton and his pledge for all americans to have a house - now called toxic mortgages and one of the precipitating factors in the current financial mess. Or Bush the 2nd and his foray into Iraq and all the joy that has brought. Or Obama and his healthcare boogie, whose constitutionality is currently being settled by the Supreme Court.
Regardless of what the talking heads on radio and TV, on either side, say the economy, particularly, is an independent beast, that can be affected by the doings of government, but for over 100 years has run in its cycles regardless of who is in office.
Fuel is an international commodity that is influenced by many players, but not the president. So the answer is expect prices to rise during the summer driving season and heating season and moderate at the rest of the year.