It depends on the item.
Rod of the Pact Keeper only increases the DC for your warlock spells:
While holding this rod, you gain a bonus to spell attack rolls and to the saving throw DCs of your warlock spells.
Rod of the Pact Keeper only raises your spell save DC when you are casting one of your warlock spells. So Rod of the Pact Keeper would not increase the DC for your familiar, unless you are casting one of your warlock spells through the familiar while using the Rod of the Pact Keeper.
For an item to work with this, it would have to read something like:
Your spell save DC increases by 1,
without specifying a use case, such as the Robe of the Archmagi:
Your spell save DC and spell attack bonus each increase by 2.
Unlike Rod of the Pact Keeper, Robe of the Archmagi does not specify when or for what your spell save DC is increased. It just increases. For everything.
For an example of another feature where Robe of the Archmagi works and Rod of the Pact Keeper does not, let us examine the Staff of Charming:
While holding this staff, you can use an action to expend 1 of its 10 charges to cast charm person, command, or comprehend languages from it using your spell save DC.
In particular, command is not a Warlock spell. So when casting command from the Staff of Charming, we would not benefit from the Rod of the Pact Keeper's bonus, since command is not one of our warlock spells, but we would get the bonus to spell save DC provided by Robe of the Archmagi.