Most markers for missions on the War Table disappear after you complete them, but some of them stick around. Are the ones that stay significant?
The ones that stay are usually one of three categories: area unlock, plot-relevant, or unfinished chain.
Area unlock missions are pretty self-evident, these missions stay as markers for you to use to fast-travel through the War Table. They are typically pyramid-shaped.
Plot-relevant missions are ones that are key to the main story and reflect your progress through the main plot. These are typically giant daggers or knives.
Some War Table missions can be chained, making the right decision of who to send can unlock more War Table missions. Thus, these missions stay on the War Table when completed, although I currently don't know whether or not them staying on the table is a sign of whether or not you unlocked the next mission. These operations are represented by tiny Inquisition emblems.