Can the person praying behind the imam read something from the Mushaf other than what the imam is reciting?

Firstly: 

What the person praying behind the imam should do is to recite al-Faatihah then listen attentively to the recitation of his imam. It is not permissible for him to recite anything more than al-Faatihah, whether he recites it from memory or from the Mushaf. 

Allaah has commanded worshippers and others to listen attentively when the Qur’aan is recited to them. Allaah says (interpretation of the meaning): 

“So, when the Qur’aan is recited, listen to it, and be silent that you may receive mercy”

[al-A’raaf 7:204]

A similar command was issued by the Prophet (peace and blessings of Allaah be upon him). It was narrated that Abu Moosa al-Ash’ari (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “The imam has been appointed to be followed, so when he says takbeer, then say takbeer, and when he recites, then listen attentively.” Narrated by Muslim, 404. 

No exception is made from that apart from the recitation of al-Faatihah only. 

It was narrated that ‘Ubaadah ibn al-Saamit (may Allaah be pleased with him) said: We were behind the Messenger of Allaah (peace and blessings of Allaah be upon him) in Fajr prayer, and the Messenger of Allaah (peace and blessings of Allaah be upon him) recited and the recitation became difficult for him. When he had finished he said: “Perhaps you were reciting behind your imam?” We said: Yes, that is so, O Messenger of Allaah. He said: “Do not do that, except for the Opening of the Book (al-Faatihah), for there is no prayer for the one who does not recite it.” Narrated by Abu Dawood, 823; classed as saheeh by Shaykh Ibn Baaz in his Fataawa, 11/221. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) said: 

It is not permissible for the one who is praying behind the imam in a prayer in which Qur’aan is recited out loud to recite anything more than al-Faatihah, rather after that he must remain silent and listen attentively to the recitation of the imam, because the Prophet (peace and blessings of Allaah be upon him) said: “Perhaps you were reciting behind your imam?” We said: Yes, that is so, O Messenger of Allaah. He said: “Do not do that, except for the Opening of the Book (al-Faatihah), for there is no prayer for the one who does not recite it.” And because Allaah says (interpretation of the meaning): “So, when the Qur’aan is recited, listen to it, and be silent that you may receive mercy”, and the Prophet (peace and blessings of Allaah be upon him) said: “When he recites, then listen attentively.” 

With regard to the exception that is made with regard to al-Faatihah only, that is because of the hadeeth quoted above, and because the Prophet (peace and blessings of Allaah be upon him) said: “There is no prayer for the one who does not recite the Opening of the Book.” Saheeh – agreed upon. End quote. 

Majmoo’ Fataawa al-Shaykh Ibn Baaz, 11/234 

Secondly: 

In the answer to question no. 52876 , it says that the person who is praying behind the imam should not hold the Mushaf behind his imam, and that holding it is contrary to the Sunnah. This applies if he is following the recitation of his imam. But if he is carrying the Mushaf and reading something other than what his imam is reciting, then this is haraam, as stated above, because it is not permissible for the person who is praying behind the imam in a prayer in which Qur’aan is recited out loud to recite anything more than al-Faatihah.

And Allaah knows best.