In the following video, Dr. Zakir Naik says that Bhishma Pitamaha tells Yudhishtira that sacrificing a cow will satisfy their dead ancestors for a period of 1 year.
Is it right? Does the Mahabharata say so?
Yes, Bhishma does say that in chapter 88 of the Anushasana Parva of the Mahabharata:
Listen to me, O Yudhishthira, what those Havis are which persons conversant with the ritual of the Sraddha regard as suitable in view of the Sraddha and what the fruits are that attach to each. With sesame seeds and rice and barley and Masha and water and roots and fruits, if given at Sraddhas, the Pitris, O king, remain gratified for the period of a month. Manu has said that if a Sraddha is performed with a copious measure of sesame, such Sraddha becomes inexhaustible. Of all kinds of food, sesame seeds are regarded as the best. With fishes offered at Sraddhas, the Pitris remain gratified for a period of two months. With mutton they remain gratified for three months and with the flesh of the hare for four. With the flesh of the goat, O king, they remain gratified for five months, with bacon for six months, and with the flesh of birds for seven. With venison obtained from those deer that are called Prishata, they remain gratified for eight months, and with that obtained from the Ruru for nine months, and with the meat of the Gavaya for ten months. With the meat of the buffalo their gratification lasts for eleven months. With beef presented at the Sraddha, their gratification, it is said, lasts for a full year. Payasa mixed with ghee is as much acceptable to the Pitris as beef. With the meat of the Vadhrinasa the gratification of the Pitris lasts for twelve years. The flesh of the rhinoceros, offered to the Pitris on the anniversaries of the lunar days on which they died, becomes inexhaustible. The potherb called Kalasaka, the petals of the Kanchana flower, and meat of the goat also, thus offered, prove inexhaustible.
Pitris are spirits of departed ancestors, by the way. In any case, see my answer here for the moral justification of animal sacrifice.
Also, it should be noted that the Brahma Vaivarta Purana says that offering meat at the Shraddha ritual is not permitted in the Kali Yuga:
In this Age of Kali, five acts are forbidden: the offering of a horse in sacrifice, the offering of a cow in sacrifice, the acceptance of the order of sannyāsa, the offering of oblations of flesh to the forefathers, and a man’s begetting children in his brother’s wife.
Now like many verses in the Brahma Vaivarta Purana, some consider this verse to be an interpolation. But this is a moot point, because pretty much no one offers meat in Shraddha nowadays; they use one of the options referred to by Bhishma that don't involve meat.
No. It doesn't support cow sacrifice though cow sacrifice is allowed during Yagnas.
मासानष्टौ पार्षतेन रौरवेण नवैव तु |
गवयस्य तु मांसेन तृप्तिः स्याद्दशमासिकी ||७||
मासानेकादश प्रीतिः पितॄणां माहिषेण तु |
गव्येन दत्ते श्राद्धे तु संवत्सरमिहोच्यते ||८||
यथा गव्यं तथा युक्तं पायसं सर्पिषा सह |
वाध्रीणसस्य मांसेन तृप्तिर्द्वादशवार्षिकी ||९||
This is how Bibek Debroy translates these verses:
With the meat of gavaya1 they are content for ten months. With the meat of buffaloes, the ancestors are delighted for eleven months. If the product from a cow2 is given at a funeral ceremony, the satisfaction is said to last for an entire year. The product from a cow can be mixed with payasam and ghee. With the flesh of vardhinasa, the satisfaction lasts for twelve years.
And footnotes say,
1 The gayal or mithun.
2 We haven’t glossed over this in the translation. The text doesn’t say meat, it says product.
Gayal or mithun is an Ox and not Gau or cow.
No, this is the wrong information given by Zakir naik. But I would like to thank him to read the Anushasana Parva of the Mahabharata or I want to give him a suggestion that he should read properly and then give any speech about our Hindu religion. Now I am showing the proper information about that chapter no 88 - 89 of the Anushasana Parva of the Mahabharata that what Bhishma tells:
The sacrifice in yajña, recommended, that is not for killing the animal, but it is a testing, how the Vedic mantras are being properly chanted. Because an old animal put into the fire, by Vedic mantras he would come out again with young life. That is sacrifice of animals in the yajña. Therefore in this age there is no such expert brāhmaṇa who can chant the mantras properly or he can behave because the life is very abominable. Therefore, because there is no expert brāhmaṇa, so these sacrifices are forbidden in this age.
In those places where it is mentioned that a cow can be sacrificed, it is actually referring to a bull, not cow per se.
In the past ages, people used to sacrifice a old bull before fire sacrifice.
They never killed a cow.
Bull is also refereed to as Go/gau etc.