PCl3 doesn't react with NEt3, although it will react with primary and secondary amines:
PCl3 + 6 RNH2 --> P(NHR)3 + 3 RNH2Cl
PCl3 is moisture-sensitive. Even the small amount of water in the air is enough.
PCl3 + 3 H20 --> H3PO3 + 3 HCl
The white fumes you saw were HCl - hope that helps.