Honestly I think the only thing that could make Bush resign are health problems of some sort. I doubt he'd resign just because the Great Recession began earlier or Iraq got worst.
Also,I feel as if Cheney might resign too and if not,he wouldn't seek his own term in office(hell,depending on when Bush resigns,Cheney might not even survive his short Presidency).
The effects could include potentially hurting the Neoconservative Movement in the Republican Party heavily,maybe taking it a bit more towards the center so McCain and Romney wouldn't have to change so much?
That or they could become more Libertarian,ala Ron Paul and Gary Johnson while also being Anti-Bush.
If Bush resigns for some reason because of the Great Recession being earlier or Iraq,I feel he would be remembered largely as a quitter or a coward by many,mainly Liberals while some Republicans would view him like that,I have a feeling a good amount of Conservatives like Frist would try and paint him as being 'forced out' by the Democratic Party.