Yes, a nonpolar molecule can contain a double bond or a triple bond.
If the molecule is a hydrocarbon, the molecule is automatically "nonpolar".
Note: these molecules will have a small polarity, because
- they are not perfectly symmetrical
- the C-H bonds have a small polarity
- some C-C bonds have a small polarity because of hybridization differences.
But the polarity differences are all so small that the molecule is still classed as nonpolar.
Compounds with Heteroatoms
Compounds with heteroatoms must be perfectly symmetrical in order to be nonpolar.
The bond dipoles must cancel for the molecule to be nonpolar.
For example, trans-1,2-dichloroethene is nonpolar but the cis isomer is polar.
Similarly, 1,2-dichloroethyne is a nonpolar molecule because it is perfectly symmetrical and all bond dipoles cancel.