Well, the first question is, why is he running again? Why isn't he, like all his predecessors except the Roosevelts, respecting Washington's two-term example? If he's the first to break with the tradition since FDR, then there's no way he wins even the Republican primary.
The major effects indeed will probably be on the primary season: first, in the absence of the 22nd Amendment, I'd imagine that it becomes tradition for the President to announce at some date whether he intends to seek re-election. This announcement is probably seen as the beginning of the primary season. Bush's announcement that he intends to seek a third term probably has the first effect of drastically changing the politics that set up the pre-loaded primary system. Which primaries move forward will probably change; the biggest change may be in Texas, where Republicans might have a reason to cooperate with Democratic efforts to move the Primary to Super Tuesday.
However, I'm not 100% that the Republican Party whole-hearted supports Bush's decision. If McCain decided to challenge Bush, he'd have to run a campaign much more similar to his 2000 run than his current one. However, Bush could probably still reach out to evangelicals, so it's not certain McCain--or a similar maverick--could defeat him. Furthermore, Bush's announcement may also induce a lot of soul-searching in the Democratic race: there's much more pressure to find a single nominee sooner, but there may also be an expectation that there's no way Bush can win again. Hence, why concede the primary when to Democrats of a certain mindset it is the election?
I'd have to imagine combination of all these forces is to strengthen the appeal of a third party candidacy, most likely Bloomberg. Though he might always take the interesting tack of running as VP to a more nationally known politician; that ticket would have Bloomberg's money (and reputation for self-sacrifice on eschewing the top of the ticket) and someone else's prestige. Not sure who he picks, perhaps Al Gore.
I must say, though, in the event of a lack of 22nd Amendment, there may not be any likely intervening butterflies until 2000: Eisenhower was getting pretty old in 1960 and isn't likely to run again. JFK probably wins since the lack of the amendment doesn't change his dad's ability to buy votes or Nixon's ability to appear impersonal. Whether JFK dies (or Watergate happens) probably depends on your personal definition of butterfly effect (are we re-running all aspects of random chance or assessing the myriad ripple effects of the POD), but assuming the later the remaining presidents are either too old or are defeated before being able to consider a third term. Until Bill Clinton, who has a great reason to run again (vindication of his policies after impeachment) and probably could have won in 2000, ceteris paribus.