I was thinking of letting my brother use my drone for a few days, but would this be covered by DJI care?  He has a recreational drone license.  

Would I get in trouble with the FAA if my brother caused property damage using my drone?

Hi Noblenutria, I see the model of your drone is Mavic Air, if your drone has purchased DJI Care Refresh, it is bound to the serial number of your drone, it can be used no matter who fly the drone, but please note it will not cover the loss or partial loss of the aircraft, gimbal or accessories.
As for whether you will get in trouble with the FAA if your brother caused property damage, please verify with FAA. Thanks.

Whoever is holding the remote is responsible.