Is there any Biblical background to making sure someone who is dating or is going to marry someone must have the father's permission of the daughter?
In the course of recording the lives of various people, the Bible does mention how men and women came together to be married. Several of these did involve the prospective groom or a representative asking a close relative of a woman for permission to marry her.
When Abraham decided that none of the women in Canaan were suitable for his Isaac's wife, he sent a servant to his homeland. There he found a suitable woman and he asked Rebekah's father and brother for permission to bring her back to Canaan to be Isaac's wife (Genesis 24:49-51).
After running away from home to avoid his twin brother's wrath, Jacob moved in with his uncle Laban. There he fell in love with his cousin Rachel. He made a deal with his uncle to work seven years for his uncle in order to marry Rachel (Genesis 29:15-20). His uncle cheated him and substituted Rachel's older sister at the marriage ceremony. Laban then offered Rachel to Jacob in exchange for an additional seven years of labor (Genesis 29:21-30). By the way, this exchange of labor for the privilege to marry did not sit well with the two sisters. Even though they loved Jacob, in their view their father Laban had sold them (Genesis 31:14-15).
Giving a daughter in exchange for labor was also done by Saul. He offered his daughter Merab to David if David would wage war on the Philistines. Saul was hoping that the Philistines would kill David, but David wasn't interested in Merab. Instead, David asked to marry his daughter Michal. Saul agreed in exchange for proof that David killed one hundred Philistines (I Samuel 18:20-29). David brought proof that he killed two hundred Philistines and so he and Michal were married.
King Solomon's marriage to Shulamith was probably arranged by her step-brothers (Song of Solomon 1:6; 8:12), but Shulamith emphasizes that she had a choice in the matter. The first three chapters of the Song of Solomon are the most complete record of a courtship in the Scriptures.
But not all marriages were arranged through the father. At one point in Israel's history civil war broke out and the tribe of Benjamin was nearly wiped out; only 600 men survived (Judges 20:46-47). Though the war ended, the Israelites had a problem. They had sworn an oath not to give their daughters to a Benjamite in marriage (Judges 21:1). The tribe of Benjamin would die out without offspring. The solution was to allow the men of Benjamin raid a gathering of virgin women. The rest of Israel would turn a blind eye to the raid; thus those of Benjamin would gain wives while the men of Israel could honestly say they did not give permission to these men (Judges 21:16-23).
Prior to the flood, men chose wives based on their beauty, which seems to be a contributing factor to the downfall of their society (Genesis 6:1-2). Hagar arranged a marriage for her son, Ishmael (Genesis 21:21). Ruel gave his daughter Zipporah to Moses as a wife (Exodus 2:16-22). Samuel picked a Philistine woman for his wife and then insisted that his parents arrange the marriage for him despite their reluctance (Judges 14:1-7). When Boaz was being dense, it was Ruth who suggested that they marry (Ruth 3:9).
The Mosaical law did not require that a father be asked to give permission to a man to marry his daughter, but it did give a father the right to halt a marriage that was forced because a man had raped an un-engaged woman (Exodus 22:16-17; Deuteronomy 22:28-29). If the marriage did take place, the man was not allowed to divorce his wife for any reason.
The main restriction that God has placed on mankind is that sexual intercourse is not to take place prior to marriage (Hebrews 13:4). In Genesis 34 is a story that sounds too much like our current society. While Dinah, a daughter of Jacob, was visiting friends, she was "discovered" by the son of the town's ruler. Shechem had sex with Dinah, whether by force or by allowance, we are not told, but we do know it was not because of love on Shechem's part. It was only after he had intercourse with her that Shechem decided he loved Dinah (Genesis 34:2-3). He asked his father to arrange a marriage, but Dinah's brothers were upset "because he had done a disgraceful thing in Israel by lying with Jacob's daughter, a thing which ought not to be done" (Genesis 34:7).
Because of the variety of methods used to find a spouse recorded in the Scriptures, we must conclude that God doesn't require any one particular method, so long as sin is not committed in obtaining a spouse. The New Testament is particularly silent on the method of gaining a spouse. It does discuss some reasons for marrying, such as the desire to have sex (I Corinthians 7:1-9), but the method used for finding a spouse is left open.
"There are three things which are too wonderful for me, four which I do not understand: The way of an eagle in the sky, the way of a serpent on a rock, the way of a ship in the middle of the sea, and the way of a man with a maid" (Proverbs 30:18-19).