In the same way that [they don't run out of ammo](https://gaming.stackexchange.com/a/54168/14), they are not affected by weight restrictions

See [here](http://masseffect.wikia.com/wiki/Combat_(Mass_Effect_3))


>**Weight Capacity**  
>Each class has a different amount of weapons that it can hold at one time. If Shepard is encumbered then limitations will be imposed. For example, power cooldowns will be extended.
>
>Weight does not seem to affect the power cooldown times of squadmates.