My Monk just hit 7th level, I'm really wanting to pick up the Greater Grapple feat, but can't until 8th level. Is there any ruling that says you must spend your feat immediately? Or can I "save" the feat slot until I hit 8th?
I don't think anything forces you to choose a feat when you hit a feat level. But even if so, as a GM I'd not only allow this, I'd probably allow you to pick one now, and retrain at 8th level.
By the rules, you cannot bank feats.
One of the primary designers mentioned that he didn't see a problem with it, but the primary designer disagreed.
Is there any ruling that says you must spend your feat immediately?
There is no rule that lets you save the feat.
When adding new levels of an existing class or adding levels of a new class (see Multiclassing, below), make sure to take the following steps in order. First, select your new class level. You must be able to qualify for this level before any of the following adjustments are made. Second, apply any ability score increases due to gaining a level. Third, integrate all of the level's class abilities and then roll for additional hit points. Finally, add new skills and feats. For more information on when you gain new feats and ability score increases, see Table: Character Advancement and Level-Dependent Bonuses.
You gain the feat when you level up. Not a generic "feat slot" to be filled at a later time.
Or can I "save" the feat slot until I hit 8th?
Officially, no you can't.
Anyhow, many GMs will allow you to either 'bank' the feat, or to pick it right away, immediately losing access to it after selecting it (kind of like a feat you lost the prerequisites to). Omce you fulfil the prerequisites, the feat comes 'online'.
Long story short: Ask your GM.