There would still be the unofficial two-term limit (or two consecutive terms anyway). FDR only broke that due to WWII. I honestly think it wouldn't change anything. Of the OTL two-termers who could have conceivably run for a third term:
- Eisenhower would respect the unofficial precedent.
- Reagan wouldn't because of his health.
- Clinton conceivably could (and might actually want to), but I doubt it. He's the most likely case though.
- Bush would be too unpopular.
- Obama conceivably could, but he almost certainly wouldn't want to.