Can We Eat Meat in al-Muharram?
Question

My husband’s family customarily avoids eating meat from the beginning of al-Muharram until the tenth of the month, and they say that this was a covenant that their forefathers took upon themselves, therefore each one of them must adhere to this covenant. Is this permissible?
Praise be to Allah.There is no monasticism in Islam 
Refraining from eating permissible things that Allah has allowed comes under the heading of monasticism that was prohibited by Allah, may He be exalted, and His Messenger (peace and blessings be upon him). 
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
“Monasticism means giving up things that are permissible, such as marriage, eating meat, and so on. A group among the Sahabah (may Allah be pleased with them) thought of following some monastic practices, but Allah, may He be exalted, sent down revelation forbidding them to do that, as He said (interpretation of the meaning): “O you who believe! Make not unlawful the Tayyibat (all that is good as regards foods, things, deeds, beliefs, persons, etc.) which Allah has made lawful to you, and transgress not. Verily, Allah does not like the transgressors” [al-Maidah 5:87]. And it is proven in as-Sahihayn that there was a group of the Companions of the Prophet (blessings and peace of Allah be upon him), one of whom said: As for me, I will fast and never not fast. Another said: As for me, I will pray qiyam (voluntary prayers at night) and never sleep. Another said: As for me, I will not marry women. Another said: As for me, I will not eat meat. The Prophet (blessings and peace of Allah be upon him) stood up to deliver a speech and said: What is the matter with men of whom one says such and such? But I fast and do not fast, I pray qiyam and I sleep, I marry women, and I eat meat. Whoever turns away from my Sunnah is not of me.” The sahih texts explain that monasticism is an innovation and misguidance.” (Al-Jawab al-Sahih (2/194-197)
To sum up: whoever seeks to worship Allah, may He be exalted, by refraining from eating meat or from other permissible things on specific days, this comes under the heading of prescribing religious practices for which Allah has not given permission.
That applies whether this is due to a belief that:

these days have some special characteristic that dictates refraining from eating meat, 
or because one regards it as haram or makruh
or because of a belief that refraining from eating meat during these days is obligatory or mustahabb
or because of a belief that one may draw closer to Allah by refraining from eating meat or from other permissible things in a particular way.

All of that comes under the heading of innovations and misguidance for which Allah has not given permission.
Ash-Shatibi (may Allah have mercy on him) said:
“Everyone who restricts himself from consuming that which Allah has permitted, without a legitimate shar‘i reason for doing so, has diverged from the Sunnah of the Prophet (blessings and peace of Allah be upon him), and the one who acts upon something other than the Sunnah by way of a religious practice is essentially an innovator.” (Al-I‘tisaam, p. 59)
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
“Allah, may He be exalted, commanded mankind to worship Him alone and not to associate anything with Him in worship, and to worship Him according to what He prescribed; and He commanded that they should not worship Him by any acts of worship other than that. Allah, may He be exalted, says (interpretation of the meaning):
“So whoever hopes for the Meeting with his Lord, let him work righteousness and associate none as a partner in the worship of his Lord” [al-Kahf 18:110]
“…that He might try you, which of you is the best in deeds” [Hud 11:7].
With regard to the one who follows the path of asceticism and worship, if he appears outwardly to be following Islamic teachings, but his intention is to show off, gain a good reputation and gain people’s respect, then his good deeds are invalid and are not acceptable to Allah. It is proven in as-Sahih that Allah says: “I am the least in need of a partner. Whoever does any deed in which he associates someone else with Me, I disavow him and all of his deeds are for the one whom he associated with Me.” And in as-Sahih it is narrated that the Prophet (blessings and peace of Allah be upon him) said: “Whoever talks about his good deeds (to show off), Allah will expose him (on the Day of Resurrection), and whoever makes a (hypocritical) display, Allah will make a display of him.”
If he is sincere in his intention, but he worships Allah by doing acts of worship other than those that are prescribed, such as one who remains silent constantly, or stands in the sun or on the roof constantly, or does not wear ordinary clothes, and always wears wool or sackcloth, and the like, or covers his face, or refrains from eating bread or milk, or drinking water, and so on – these acts of worship are invalid and are to be rejected, as it is proven in as-Sahih in a report from ‘Aishah that the Prophet (blessings and peace of Allah be upon him) said: “Whoever introduces anything into this matter of ours that is not part of it will have it rejected.
And it is proven from him in as-Sahih that there was a group among his Companions, one of whom said: As for me, I will fast and never not fast. Another said: As for me, I will pray qiyam (voluntary prayers at night) and never sleep. Another said: As for me, I will not marry women. Another said: As for me, I will not eat meat. The Prophet (blessings and peace of Allah be upon him) stood up to deliver a speech and said: What is the matter with men of whom one says such and such? But I fast and do not fast, I pray qiyam and I sleep, I marry women and I eat meat. Whoever turns away from my Sunnah is not of me.”
So this is the principle with regard to acts of worship, because fasting and prayer are both acts of worship.
Refraining from eating meat or getting married is permissible in principle, but when that went beyond the framework of the Sunnah and the individual committed himself to more than is prescribed, or committed himself to refraining from that which is permissible, as monks do, the Prophet (blessings and peace of Allah be upon him) disavowed those who do such things as having turned away from his Sunnah towards something else, and he said: “There is no monasticism in Islam.” (Majmu‘ al-Fatawa (11/612-614)
Can we eat meat in Muharram?
Having the custom of not eating meat during Muharram, in the manner mentioned – whether that was based on a covenant with Allah, or with other people, or with one’s own self, or without any covenant – all comes under the heading of innovation that is not permissible as a means of drawing closer to Allah, may He be exalted, and it is not permissible to obey one’s parents and grandparents concerning that, or to follow them in doing it.  
For more about al-Muharram, , please see these answers: 193281 , 204142 , 21311 , and 128423 .     
For more about monasticism, please see these answers: 6254 , 34652 , and 21942 .
And Allah knows best.