In part two of Fahrenheit 451, what homemade communication tool did Faber give Montag?

In Part Two, Montag visits Faber's home in hopes of learning how to comprehend the information he has been reading. Faber is initially reluctant to help Montag challenge the oppressive authoritarian regime, but he eventually agrees and offers him a homemade communication device known as the green bullet. The green bullet resembles a tiny Seashell ear radio but is a two-way communication device, which will allow Montag to speak with Faber imperceptibly. Faber mentions that when he had made the two-way communication, he planned on resisting the government but was too cowardly to act.

Montag takes the green bullet and Faber reads the Bible to him throughout the night. Faber also gives Montag valuable advice via the green bullet while Captain Beatty attempts to dissuade him from reading books. Eventually, Captain Beatty notices the green bullet and hits it out of Montag's ear. When Beatty threatens to locate the person on the other end communicating with Montag, Montag shoots and kills him using a flamethrower.

In Part Two of “Fahrenheit 451” called “The Sieve and the Sand”, Faber gives Montag a homemade device that they will be able to use to communicate with one another.  This device is a two-way radio that Montag will put into his ears so that he and Faber will be in continuous communication.  This two-way radio is used throughout the novel for the two men to have easy, private, and safe communication with one another.  It is not difficult for the men to hide the fact that they are communicating through these since other characters use them as a way to listen to their televisions and hear constant sound – Milred uses them in the novel.

What device did Faber design to stay in constant contact with Montag?

Faber has invented a type of radio that looks like the Seashell Radio that Mildred and many others keep in their ears. 

When Montag goes to Faber's house, seeking help to understand what it is that books have to offer, Faber reluctantly agrees, but he gives Montag a device which fits in the ear. Using this device Montag can hear Faber as if he were in a room with him. Having lived alone for years, Faber has toyed with electronics until he has succeeded in creating a transmitter and a receiver so that he can be in two-way contact with another person. He displays his listening device to Montag: 

He [Faber] picked up a small green metal object no larger than a .22 bullet.

Faber tells Montag that he can sit in his home and listen in on the firemen:

"Eventually, I could put out ears into all parts of the city, with various men, listening and evaluating."

Faber apologizes for being cowardly as he sits in the safety of his home, but he agrees to transmit advice and encouragement to Montag if the fireman will keep the "bullet" in his ear. So, as the narrative develops Montag keeps this bullet in his ear and Faber advises him when Montag mistakenly thinks the friends of Millie will appreciate his reading of "Dover Beach." He tells Montag to throw the book away. Faber also talks to Montag when in the company of Beatty, but, unfortunately, Beatty discovers it. 

What is the device invented by Faber in Fahrenheit 451 and its intended use by Montag?

I assume that you are referring to the device that Faber gives Montag to put in his ear.

The device (referred to as "the bullet") is about the size and shape of a .22 caliber bullet.  It fits in the ear and people will not necessarily know it's there.  It is a two-way communications device.  Faber presumably has some sort of device in his house that communicates with the bullet.  It lets him speak to Montag and it lets him hear whatever is going on around Montag.

Montag is going to use the device to help subvert the society he lives in.  Faber will, for example, tell him what to say to people.  He will also talk to Montag to help him understand and resist things that Beatty says to him (that are meant to force Montag to conform).  So Faber will kind of talk Montag through his days, helping him to try to make people doubt the society they live in.

What is the device Faber gave Montag for communication in Fahrenheit 451?

In Part Two "The Sieve and the Sand" of the novel Fahrenheit 451, Montag travels to Faber's house trying to find meaning in the texts he is reading. Faber is hesitant to help Montag because he fears that the government will arrest him. Montag suggests that they make copies of books and distribute them. Then, Faber makes the suggestion that they should "plant" the books in the homes of firemen to destroy the entire system. Faber tells Montag he is only joking, and then admits to him that he is a complete coward. Before Montag leaves, he asks Faber if there is any way he can help him with Captain Beatty. Faber opens his bedroom door and leads Montag down a small hallway to a room with tools, gadgets, and various mechanical devices. Faber gives Montag his two-way listening device called the "green bullet." The green bullet resembles a Seashell radio and allows Faber to not only communicate with Montag but also listen to his conversations. Later that night, Faber reads Montag the book of Job via the green bullet.