Do you have to read the first Wheel of Time books before reading the Mistborn trilogy?

These published volumes are not connected to the same storyline, and can be read separately.

Brandon Sanderson is an author in his own right, and the Mistborn trilogy was written by him alone and can be read as a single trilogy. Mr. Sanderson was selected by Robert Jordan, the author of the Wheel of Time series, to collaborate on and write the last volumes of the series because Mr. Jordan was ill and died before he could complete the series. It should be noted that Mr. Jordan always maintained he knew when he began writing the first volume exactly how the Wheel of Time series would ultimately end.

Mistborn

1. The Final Empire (2006)

2. The Well of Ascension (2007)

3. The Hero of Ages (2008)

4. The Alloy of Law (2011)

Wheel of Time

1. The Eye of the World (1990)

2. The Great Hunt (1990)

3. The Dragon Reborn (1991)

4. The Shadow Rising (1992)

5. The Fires of Heaven (1993)

6. Lord of Chaos (1994)

7. A Crown of Swords (1996)

8. The Path of Daggers (1998)

9. Winter's Heart (2000)

10. Crossroads of Twilight (2002)

11. Knife of Dreams (2005)

12. The Gathering Storm (2009) (with Brandon Sanderson)

13. Towers of Midnight (2010) (with Brandon Sanderson)

14. A Memory of Light (2013) (with Brandon Sanderson)

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Just a correction to what "S" said, Robert Jordan did not chose Sanderson to continue the Wheel of Time, Harriet (Robert Jordan's Widow) did.

Otherwise what he said was pretty much spot on, the two series are as related as Lord of the Rings and Harry Potter. Meaning their only relation is the are categorized as fantasy.