Mormons believe both passages,
That Adam and Eve were commanded to multiply (Genesis 1:28).
Unless these partook of the fruit of the tree of knowledge Adam and Eve could not have children (2 Nephi 2:23).
- God commanded them not to partake of the tree of knowledge of good and evil (Genesis 2:17).
So really, the contradiction in Mormon belief seems to be in the commandments God gives to them: multiply and don't eat...but you can't multiply until you eat.
God is forcing them into a situation in which they cannot follow his commandments. Is God the author of sin?
Other Christians don't face this contradiction because #2 is unique to Mormonism.
AFAIK, there isn't a direct, clear-cut answer to this in canon, but here are two scripturally-supported common explanations.
Explanation 1: It was a transgression, not a sin
Mormons consider transgressions and sins differently.
A transgression is violating a law. A sin is violating a law, knowing it to be evil.
Joseph Fielding Smith, a President of the LDS Church said
I never speak of the part Eve took in this fall as a sin, nor do I accuse Adam of a sin...This was a transgression of the law, but not a sin...for it was something that Adam and Eve had to do!
Just as Mormons believe little children cannot sin (Moroni 8:8) they believe that Adam and Eve could not sin, as they did not know good from evil.
Thus, God's commandments were not a sadistic trap that forced them to sin. Instead, because they could not sin, he was offering a sin-free choice: live innocently in the garden forever without joy or pain, or leave the garden, know good and evil, and have posterity.
Eve chose the latter and Adam followed.
Explanation 2: One commandment superceded the other (for now)
Mankind (according to Mormonism and many other Christian faiths) is still under the command to "multiply and replenish the earth".
However, my adult sister is not participating in this commandment. Why? Though she could physically bear children, she is not married, and God has forbidden adultery/fornication. So on the one hand, God commands her to multiply, but on the other hand, another of God's commandments prevents her, at least for now.
This is the situation that Adam and Eve faced. They were to multiply, but another commandment (temporarily) restricted them from doing that. Had Adam and Eve refrained from partaking of the fruit, circumstances would have changed, i.e. God lifts the restriction, they fulfill his commandment to multiply, without repercussion.
Of course, Adam and Eve chose to take the fruit anyway, and God knew they would, but God wasn't contradicting himself then, just as he isn't contradicting himself now.
It is worth noting that rather than viewing the Fall of Adam and Eve as an unfortunate and evil accident, one which if it had not happened, everything would be perfect, Mormons see the Fall necessary to the purpose of life on earth.
Life is an opportunity to learn to choose good over evil. This could not happen if there was not good or evil. (See 2 Nephi 2.)
The sacrifice of the Son of God was planned from the beginning:
the book of life of the Lamb slain from the foundation of the world
So also was the Fall -- which necessitated the sacrifice -- known from the beginning. Mormons often describe the Fall as a fall downward, but also forward.