Does The Bible Say That We Should Cry At The Birth Of A Child And Rejoice At The Death Of A Saint?

No, this is not from the Bible. However, there are times in the Bible when the speaker suggests that one should weep at birth and rejoice in death. Job, I think, speaks in that way, cursing the hour of his birth and wishing that death would claim him and end his misery. I think (without having researched) that some of the prophets speak of a time when things will be so bad for the nation of Israel that mothers will cry when their children are born and be envious of those who die, ending their suffering. But most of it is contextual and not an actual instruction. For the most part, the Bible is very positive about childbirth.

This is not in the Bible, but is a Jewish or Irish or Buddhist saying. Here is a link to a pastor's blog who explains it pretty well. xrysostom.blogspot.com

That used to be said in that people considered death to be going home to be with God and have no more hardships.  so people should be glad for them.. And to cry at birth because it is the beginning of life and hardship> I haven't heard the expression in a long time.  It was something from my grandmothers day I think from the early 1900's

Well not exactly....The actual verse is at Eccleciastes 7:1, but verses 2and 3 gives some perspective as well.
As part of Solomon's expository on the meaning and purpose of life he emphasizes here that quiet reflection is more profitable the being in a constant state of mindless jubillance (see Eccl 2:2). The Epicurean attitude that is so prevalent now is discouraged by one who could afford to experience it all. It is thus important to consider the purpose of life as stated at Eccl. 12:13,14- "....fear the true God and keep his commandments. For this is the whole obligation of man....".

Because a newborn has to live out an entire life filled with sin, pain, and suffering whereas a dead saint has earned their heavenly reward, and their Earthly hardships are over.

I didn't actually know it was an excerpt from the bible. But I often heard it said that you should cry at a birth and laugh at a death, meaning that when we are born we have to suffer the trials and tribulations that life throws at us. But in death we are at eternal rest enjoying the company of god. I always thought it was just a saying.

And here is the verse! It is indeed in the bible
Ecclesiastes 7:1-4 

A good name is better than precious ointment, and the day of death than the day of birth. It is better to go to the house of mourning than to go to the house of feasting, for this is the end of all mankind, and the living will lay it to heart. Sorrow is better than laughter, for by sadness of face the heart is made glad. The heart of the wise is in the house of mourning, but the heart of fools is in the house of mirth.

There were no saints in the Bible, so no.

Genesis 3:16 - "To the woman He said, "I will greatly multiply Your pain in childbirth" this was a punishment for Eve's sin.

The only occasion celebrating death I am aware of is Good Friday where we celebrate Christ's sacrifice on the cross. This is a Christian tradition and not a biblical holy day/feast day.