Does one sin if he says to himself, "What does the Messenger of Allah (peace be upon him) have to do with it?" upon hearing someone say, "We put our trust in the Prophet of Allah," and does that apostatize him from the religion?

What you said is correct, and there is no disparagement of the Prophet (peace be upon him) in it. Reliance (Tawakkul) is an exclusive attribute of divinity and it is not permissible to direct it to other than Allah. You should advise this person gently and kindly, explaining that reliance is an act of worship that is directed only to Allah, and reliance on other than Allah is polytheism (shirk). Allah has singled Himself out for reliance and desire in His saying: "And if only they had been satisfied with what Allah and His Messenger gave them and said, 'Sufficient for us is Allah; Allah will give us of His bounty, and [so will] His Messenger. Indeed, to Allah we aspire.'" [At-Tawbah: 59]. And in Al-Fatihah, we recite: "It is You we worship and You we ask for help," and the precedence of the object (You) indicates exclusivity, meaning that seeking help and reliance can only be with Allah. This is further supported by the Prophet's (peace be upon him) saying: "And if you seek help, then seek help from Allah."