Firstly: 

Allah has prescribed fasting during the month of Ramadhan so that those who fast may attain taqwa (piety). Allah says (interpretation of the meaning): 

“O you who believe! Observing As-Sawm (the fasting) is prescribed for you as it was prescribed for those before you, that you may become Al-Muttaqoon (the pious)” [2:183] 

If that colleague who eats the flesh of people (by gossiping) does not pay heed to this month then when will he pay heed, repent and fear Allah? 

It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “Whoever does not give up false speech and acting upon it, Allah has no need of him giving up his food and drink.” [al-Bukhaari]

‘Umar ibn al-Khattab (may Allah be pleased with him) said: “Fasting is not just (giving up) food and drink, rather it is giving up lying, falsehood and idle talk.” 

Jabir ibn ‘Abd-Allah al-Ansari (may Allah be pleased with him) said: “When you fast, then let your hearing, your sight and your tongue fast from lying and sin, stop abusing servants and be tranquil and dignified on the day when you fast. Do not let the day when you do not fast and the day when you fast be the same.” 

Those who backbite about people and “eat their flesh” should beware lest their fast be lost and lest all they get from it is hunger and thirst. 

It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “There may be a fasting person who gets nothing more from his fast that hunger and thirst, and there may be a person who prays qiyam and all he gets from his qiyam is a sleepless night.” [Ahmad]. 

Some of the scholars are of the view that committing sins invalidates the fast. 

Hafsah bint Sireen (may Allah have mercy on her) said: “Fasting is a shield, so long as the one who fasts does not break it, and what breaks it is backbiting.” 

It was narrated that Ibrahim al-Nakha’i (may Allah have mercy on him) said: They used to say that lying breaks the fast of one who is fasting. 

This was also the view of some of the salaf (pious predecessors), which is that all sins break the fast, and whoever commits a sin when he is fasting has to make it up. This is the view of al-Imam al-Awza’i, and was the view favoured by Ibn Hazm al-Zahiri (may Allah have mercy on him). 

But the majority of scholars are of the view that committing sins reduces the reward for fasting, but it does not invalidate the fast, and this is the correct view. See the answer to question no. 50063. 

Secondly: 

Backbiting is a major sin. Allah, may He be glorified and exalted, says (interpretation of the meaning): 

“O you who believe! Avoid much suspicion; indeed some suspicions are sins. And spy not, neither backbite one another. Would one of you like to eat the flesh of his dead brother? You would hate it (so hate backbiting). And fear Allah. Verily, Allah is the One Who forgives and accepts repentance, Most Merciful”

[49:12] 

It was narrated that Anas ibn Malik said: The Messenger of Allah (peace and blessings of Allah be upon him) said: “When my Lord took me up into heaven, I passed by some people who had nails of copper with which they were scratching their faces and chests. I said: ‘Who are these, O Jibreel?’ He said: ‘These are the ones who used to eat the flesh of the people and impugn their honour.’” [Abu Dawood]

Thirdly: 

You have to denounce this colleague of yours and not approve of what he is doing. 

It was narrated that Abu Sa’eed al-Khudri (may Allah be pleased with him) said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say: “Whoever among you sees an evil action, let him change it with his hand (by taking action); if he cannot, then with his tongue (by speaking out); and if he cannot, then with his heart (by hating it and feeling it is wrong), and that is the weakest of faith.” 

Al-Nawawi (may Allah have mercy on him) said: 

Note that the one who hears any backbiting about a Muslim should refute it and rebuke the one who says it. If he does not rebuke him in words he should rebuke him by his actions. If he cannot rebuke him by his actions or his words, then he should leave that gathering. If he hears any backbiting about his Shaykh (teacher) or anyone else who has a right over him or is one of the righteous and virtuous people, then he should pay even more attention to what we have said. 

In the book of al-Tirmidhi we narrated from Abu’l-Darda (may Allah be pleased with him) that the Prophet (peace and blessings of Allah be upon him) said: “Whoever protects the honour of his brother, Allah will protect his face from the Fire on the Day of Resurrection.” [Al-Tirmidhi] End quote. Al-Adhkaar. 

Shaykh ‘Abd al-‘Azeez ibn Baz (may Allah have mercy on him) was asked: 

I am a young woman who hates backbiting and gossip. Sometimes I am with a group of people who talk about others and they start backbiting and gossiping. In my heart I hate and loathe this, but because I am very shy I cannot tell them not to do that, and there is no place I can go to get away from them. Allah knows that I wish that they would talk about something else. Is there any sin on me for sitting with them? What should I do? May Allah help you to do what is best for Islam and the Muslims. 

He replied: 

There is sin on you for that unless you denounce the evil. If they accept that from you, then praise be to Allah, otherwise you have to leave them and not sit with them, because Allah, may He be exalted, says (interpretation of the meaning): 

“And when you (Muhammad) see those who engage in a false conversation about Our Verses (of the Quran) by mocking at them, stay away from them till they turn to another topic. And if Shaytan (Satan) causes you to forget, then after the remembrance sit not you in the company of those people who are the Zalimoon (polytheists and wrongdoers)” [6:68] 

“And it has already been revealed to you in the Book (this Quran) that when you hear the Verses of Allah being denied and mocked at, then sit not with them, until they engage in a talk other than that; (but if you stayed with them) certainly in that case you would be like them”

[4:140] 

And the Prophet (peace and blessings of Allah be upon him) said: “Whoever among you sees an evil action, let him change it with his hand (by taking action); if he cannot, then with his tongue (by speaking out); and if he cannot, then with his heart (by hating it and feeling it is wrong), and that is the weakest of faith.” [Muslim]. 

And there are many other verses and ahadeeth (prophetic narrations) which say the same thing. And Allah is the Source of strength. End quote. 

Majmoo’ Fatawa al-Shaykh Ibn Baz. 

So strive to remind your colleague of the ruling on backbiting, and explain to him the punishment of the one who does that. Perhaps he will give up what he is doing of sin. Remind him that backbiting during Ramadhan is more sinful, and the basic principle is that you should avoid sitting with him if he persists in not responding to the command of Allah. But seeing as you work with him and you cannot leave the place, then avoid listening to him and stop paying any attention to what he says. You can use the means of threatening to report him to his superiors at work, or threaten to tell the people about whom he is talking. Perhaps if he does not fear Allah he will fear people and will stop backbiting about them, and you will be relieved of having to listen to his annoying talk. 

And Allah knows best.