iserith
Magic Wordsmith
While passive Investigation is not specifically called out in the final rules, there is a sidebar indicating that just about any skill can be used passively by taking the modifier and adding 10.
For my purposes, I tend to only keep track of the PCs' passive Insight and Perception scores, as those seem to me to be the two skills that are more passive than active anyway. I sometimes go with passive knowledge scores, and I've also started using passive Initiative in one of my games. But I've yet to actually use passive Investigation.
My issue with the Observant feat is that it doesn't really make sense to me that someone would be better at noticing things when they're *not* actively looking for them. Surely the feat really ought to grant advantage on active checks as well as the +5 advantage-equivalent to the passive scores.
Consider that "passive" refers to the fact that the player is not rolling any dice, not that the character is not doing a thing actively. If you take a look at the rules on passive checks, you'll see that's what it is referring to. Passive checks are used when the DM wants to make a check secretly and/or for when the DM wants to resolve uncertainty in a task that is being performed repeatedly.
In the case of passive Investigation, a character might be trying to deduce the significance of the faded hieroglyphs on the walls of the dungeon as the party travels its halls and chambers. (That the hieroglyphs are faded and the characters are traveling is perhaps what makes the outcome uncertain.) If the character's passive Investigation is high enough, then the character is able to arrive at a chilling deduction: That these hieroglyphs warn of a terrible curse that will befall any who desecrate the inner sanctum of the dungeon at the threshold of which the party now stands.
Unlike passive Perception, passive Investigation is probably not likely to come up a bunch, though of course that depends on the DM.