A new council takes its place, with virtually no effect on the story.
It does affect a Codex entry and dialogue with some NPCs on the Citadel in ME2, and it is a bit more difficult to get your Spectre status restored (utterly trivial, given that it's barely acknowledged and you get reinstated in ME3 regardless). In ME3, it strengthens some War Assets while locking you out of others, but its main impact is that it raises the difficulty of the Charm/Intimidate check in the standoff with Kaidan/Ashley, which could result in the loss of a potential squadmate. The new Councilors are quite different from their predecessors in terms of personality, but you don't interact with them much anyway.
In short, there's a lot of minor differences, but it doesn't affect the critcal path at all. Terminator-HIX