No, domination points are not increased/decreased during a person's turn. If they suicide and flag the same turn, no increase in the flag cost even if they moved up a place or two just before flagging.

However, a suicide may work in a situation past round 3 where the player was last/second to last to originally act (just before the round change). In theory, if they suicide and hold a high enough place as the round changes over, they would gain dom points and be able to flag for a little less.