Mitt Romney's resident state is Massachusetts and his birth state is Michigan. In the 2008 presidential election both Michigan and Massachusetts both voted for Barack Obama for the presidency in overwhelming fashion. In fact, both Michigan and Massachusetts have voted for the Democratic candidate for president in every election since 1992. Romney faces an uphill battle for those states. He will probably not win those states unless something crazy happens from now until November. While Michigan has had closer races in both 2004 and 2000, the Democratic candidate still received 51% of the votes in each of those elections. It's been since 1992 that the Democratic candidate has not received at least 60% in the election in Massachusetts.
There has not been a president who has lost both his birth and his resident state. There have been presidents who have lost either their birth state or their resident state but none of them have ever lost both. George W. Bush is the most recent example, he lost Connecticut in both 2000 and 2004 but won Texas both years, as well as the presidential election nationally. George McGovern, a Democrat from South Dakota, was the last presidential candidate to lose his birth and residential state. Granted, they were the same, but still. You would have to go back to Adlai Stevenson in 1952 and 1956 to find a presidential candidate who had two separate states for birth and residential states that lost both of them.
It's possible for Romney to become president while losing both his birth and residential states but it seems unlikely. It would be the first time in history.
Update: James Polk was elected president when he lost his birth state of North Carolina and his residential state of Tennessee. It would be the second time in history if Mitt Romney was able to win the election despite losing both his birth and residential state.