If your adding a unit, civ, building or improvement etc
Wrong - all of those can affect saved games, so should have the box ticked.
Adding a unit without affects save games checked - any player builds the new unit, save game, exit, restart without mod, reload game. The already built units will at best adopt the stats of the "next" unit in the database, at worst will crash the game. I had one memorable CivV game where all my replacement swordsmen turned into super-carriers as the game does not store unit type but unit id references.
Same is true for buildings. Improvements will usually just crash the game, especially if a worker was in the process of making one when the save happened.
Adding a civ without affects save games checked - playing a game where either the human or the AI is that civ, save and exit, restarting without the mod will almost always crash the game as so much data is missing from the database that the .civbesave file refers to.
However, not all database updates need the box checking. Consider a mod that only changes the max XP for aliens from 30 (2 promotions) to 100 (4 promotions). This does not affect saved games. If I start with this mod enabled, and go on an alien extinction spree, lots of my units will get XP over the default 30. I then save, exit, restart without the mod. All my units still have the XP previously gained, they don't magically lose it as the game has no memory of where the XP came from, just how much XP each unit already has. If I continue the alien extinction spree all that happens is my units will not get any XP for aliens above the original 30.
The same is true if I change a prereq tech - if that's all the mod does I do not need to check the box. With the mod loaded I need tech Z to build the units, but once built the unit no longer cares (ie doesn't store any data about) what the prereq tech was. If I save, exit and restart without the mod, I'll still have all those units, but if I want to build another I'll need to have researched tech X not tech Z. The only issue will be if a city was in the process of building a unit when the save occurred. There are two possibilities, a) the game doesn't recheck the prereq tech on load, so the unit will finish being built, or b) the game does check and the unit will be removed from the build queue losing any prod so far - annoying but not game breaking.
Stats are difficult to predict. For example, cost doesn't affect saved games - things built will remain built, future builds will take the new cost, things at the top of the build queue will either complete at the old price (unlikely) or complete at the new price (how CivV works). However, some building stats are added to the city (eg those that add HP) so you will get weird results when the building is destroyed and the different stat is used to remove the value from the city as they will be different - eg new command centre adds 75HP (vs 50 for old), city has old built, save, reload with mod adding new values, CC is destroyed, city will have 100 + 50 - 75 = 75 HP; do it the other way around and the city will have 100 + 75 - 50 = 125 HP. Neither is game breaking (unless the final result is negative HP for a city), but both will confuse players.
Short answer - if your mod adds to existing primary database tables, check the box; if your mod updates the database you'll need to think it through; if your mod is only a UI mod, uncheck the box; if in doubt, check the box.