It would set a precedent and become standard. Once elected a president would be expected to keep running for re-election until he is either voted out or dies. Term limits might still be put in place later on, but we might have a four or five term limit, rather than two.
Additionally, it will also become the precedent to follw in other emerging democracies in the 19th century. In Latin America, for example, folks like Porfirio Diaz would be quite happy to have this precedents established for them. You could also expect quite a few republics world wide to become quasi-monarchies as the President in power can always boost up his son politically to run for office after he dies; the Lopez dynasty in Paraguay might last longer (although their downfall was the result of their defeat in the War of The Triple Alliance then their dynastic rule so it might not be the case here).
Another side effect that might have happened, is that the office of the Presidency devolves into mostly a figure-head position, with the Speaker of the House having a much more active prime-minister-like role. If this becomes the case, it would be quite rare to vote out a President, and it would likely always happen in tandem with the speaker and a big shift within the government.