In a short answer No. the main resrictions are that they must be over 35 and beborn in the US...this stopped Henry Kissenger from running (he was born in germany).
sft42's point about having to be born in the US is also the reason why Tony Blair can only dream of being President of the USA (I know a few Americans who dream of him being their Prez too...). Ironically, while he would be able to run as Vice President if he became a US citizen for a number of years first (25, as I recall) he wouldn't automatically succeed to the Presidency if the incumbant died in office.