The argument around whether Paul is truly infected or not truly differs depending on who you talk to. I lean mostly towards the fact that he ISN'T infected, and I'll tell you why!
These lyrics from 'Inevitable' at the end of the musical:
"What if I told you a story
That settled all the dust
I'm still the man you trust"
"What if the only choice is
You have to sing to survive
We must go on with the show"
I believe these can be interpreted as "Emma, please, you have to sing to survive, we have to escape this and this is the only way how!"
Not only that, but Paul then proceeds to sing all the songs he heard throughout the day such as "Get Your Cup of Roasted Coffee", "Show Me Your Hands", "Not Your Seed", and "Show Stopping Number". Now, for that last song, he wasn't present for most of the song but he did arrive at the end to rescue Ted and Emma.
He wasn't present for "America Is Great Again" or "Let Him Come" so he didn't sing them, now the only song he sung but he wasn't there to listen to was "You Tied Up My Heart". Now he could have either heard it, as they were all in the same complex or he could have heard the other infected people singing it. It's unlikely, as he would've alerted the group of Sam's singing and the other infected people wouldn't be singing it since it was specifically for Charlotte and Sam. But then again, they ARE a hive-mind and probably know the song from their near telepathic style of communication.
So what do you think? Is Paul really infected?