(Answer updated since I've largely finished the game)
There are many places where your progress in the main game (here defined as quests that existed before Phantom Liberty) affect quests in Phantom Liberty. Some examples are:
In the Phantom Liberty epilogue, what you did with Delamain in Don't Lose Your Mind affects the conversation. The list of netrunners you can call in Black Steel in the Hour of Chaos varies on your progress in the main game, and whether the people you interacted with survived. There are more options in Balls to the Wall if you've already met Rogue, River, or Panam and completed their side jobs.
On the other hand I found only a few examples of extra dialogue where Phantom Liberty progress affects the main game.
If you've already completed You Know My Name prior to meeting Lizzy Wizzy in Violence, you will comment on how you've already met before, and she will say she does not want to publicize her visit to the Black Sapphire. (Conversely, if you've already completed Violence prior to starting You Know My Name, she will give you an iconic headgear in the Black Sapphire.) Aside from that, you can mention Songbird to Alt Cunningham when you meet her in the net, during We Gotta Live Together.
And of course,
Nocturne Op55N1 is affected. If you've completed Phantom Liberty and unlocked Who Wants to Live Forever, then you can call Reed from Misty's Esoterica.
A few other things changed. You get extra random encounters, and quite a few gigs now have extra car chase stages. The v2.1 quest I Really Want to Stay at Your House is borderline - you can ask your partner to Dogtown - but they will refuse in a text message. So: aside from the instances above, there are no extra lines of spoken dialogue. Or put differently: if you are aiming to maximize your options, do the main game before Phantom Liberty.*
*Caveat to this: depending on your choices at the end of Phantom Liberty, you can get some unique iconic items that can't be gotten anywhere else. If you do the main game before Phantom Liberty, then there's not much game left to use these items on.