Was there any promise of eternal life associated with any of the
Mosaic Law. For instance, was animal sacrifice promised with eternal
No. Eternal life is foretold in the Old Testament, but never as a promise associated to obedience to Mosaic Law.
The following verses refer to life after death, hinting at the eternal nature of life. This explains the how the man in Matthew 19:16 knew about eternal life.
All are NIV.
Though you have made me see troubles, many and bitter, you will
restore my life again; from the depths of the earth you will again
bring me up.
But your dead will live; their bodies will rise. You who dwell in the
dust, wake up and shout for joy. Your dew is like the dew of the
morning; the earth will give birth to her dead.
12 Therefore prophesy and say to them: ‘This is what the Sovereign
Lord says: My people, I am going to open your graves and bring you up
from them; I will bring you back to the land of Israel. 13 Then you,
my people, will know that I am the Lord, when I open your graves and
bring you up from them. 14 I will put my Spirit in you and you will
live, and I will settle you in your own land. Then you will know that
I the Lord have spoken, and I have done it, declares the Lord.’”
And one that clearly teaches everlasting life as part and parcel with resurrection:
Multitudes who sleep in the dust of the earth will awake: some to
everlasting life, others to shame and everlasting contempt.
However, these are prophetic or poetic works, not books of the Law. I'm not aware of any teachings on eternal life in the Pentateuch - this first five books, which are commonly referred to as the "books of the Law". Technically, those are "The Law of Moses" or "Mosaic Law", so it would seem that the answer to your question is "no" if you wish to be that precise, but there are Old Testament teachings/revelations/writings on it.
From a traditional Sola Fide perspective, you wouldn't expect it to, though.
Most of the Mosaic laws talked about consequences or punishment for sin, along with a few promises of reward for obedience.
However, Scripture clearly teaches that eternal life is not earned through obedience to the Law. It is a gift from God, not of works. (Ephesians 2:9.) We are not justified by works of the law (Galatians 2:16)
If the Mosaic Law promised eternal life based on obedience to the Law, the entire message of Christianity would be invalidated.