At least some of the plot emphasis probably reflects the underlying text: really, the text only mentions the coat twice: first, when Jacob gave it to him (Genesis 37:3), secondly, when Joseph visits his brothers while wearing it (v. 23), and third when Joseph's brothers sell him into slavery, dye the tunic with goat's blood, and bring it to their father as "evidence" that he was killed by wild animals (v. 31 - 33).
The primary reason that the text even mentions it in the first place is as an illustration of Jacob's blatant favoritism (which was a major factor in his brothers' jealousy), and so that the brothers could use it later to fake Joseph's death:
Now Israel loved Joseph more than all his sons, because he was the son of his old age; and he made him a varicolored tunic. (Genesis 37:3, NASB)
Later, when Joseph shows up wearing it to check on his brothers, they were finally so resentful by his (perceived) arrogant behavior that they decide to kill him.
“Here comes that dreamer!” they said to each other. “Come now, let’s kill him and throw him into one of these cisterns and say that a ferocious animal devoured him. Then we’ll see what comes of his dreams.” (21-22)
It's only due to the dissuasion of Ruben (the oldest) that they refrain from killing him. Instead, Judah convinced them (while Ruben was gone) that it would be pointless to kill him and that they should sell him into slavery instead. At this point, they decide to fake his death and make it look like an accident:
31 Then they got Joseph’s robe, slaughtered a goat and dipped the robe in the blood. 32 They took the ornate robe back to their father and said, “We found this. Examine it to see whether it is your son’s robe.”
33 He recognized it and said, “It is my son’s robe! Some ferocious animal has devoured him. Joseph has surely been torn to pieces.”
So, then, (as previously mentioned) the cloak served two purposes in the text: as a symbol of Jacob's favoritism and as a means for the brothers to cover up the fact that they sold him into slavery.
An interesting side note: some translations don't even refer to it as "multicolored" - for example, the translators of the New International Version, feeling that the word was somewhat ambiguous, merely referred to it as an "ornate" tunic without any mention of it being multicolored.
The significance of the coat, then, is primarily that it's the instigator of the story. It's symbolic of Jacob's favoritism, the favoritism that set the events of the story into motion, and is the means by which the brothers sell him into slavery and make it look like Joseph was killed by wild animals.
The musical, of course, adds the symbolism of Joseph wearing the coat again when he was reunited with his father. (The text doesn't actually mention anything like this occuring, but it's good symbolism for the musical). In the musical, it's symbolic of loss and regain - it's both something that (indirectly) instigates the story and something that's there at the end of the story as well.
In the original text, it's only actually essential to the plot itself at two points: the point where Jacob gave it to him (as a symbol of his favoritism) and the point at which the brothers sell him into slavery. The musical take it one step further and uses it as a metaphor for the entire storyline as well as the theme of loss and regain.