Since DBs do not reuse numbers of deleted records it is possible to run out of numbers, especially if you pick not really a big integer type for this column.
What would
For MySQL, it is documented that:
The behavior of the auto-increment mechanism is not defined if a user assigns a negative value to the column or if the value becomes bigger than the maximum integer that can be stored in the specified integer type.
In Postgres, the "serial" type is equivalent to the creation of a SEQUENCE with the NO CYCLE option, and setting the default of the field to nextval. Exhausting such a sequence produces an error:
http://www.postgresql.org/docs/8.3/interactive/sql-createsequence.html
It depends on your database, I believe in MS SqlServer, you simply cannot insert any new rows until you fix the problem. The last time I encountered it, we fixed the problem by reseeding the identity column to 1. That's obviously not a universal fix, but it was ok with our situation.
I think exactly what happens will be dependent on which database engine you're using (there may even be differences between INNODB and MyISAM in MySQL). Whatever happens, it's not going to be pretty.
You'd simply have to change the column type to a larger integer.