You could use either of those and you would be understood.

Technically "do you work or study?" would be most commonly used if you don't know the person and have no idea what they do.

"Are you working or studying?" would generally be used if you are familiar with the person but aren't sure what they are doing at the moment.

I hope this helps :)

hi, i am studying as well as  earning. I am crazy about learning english,i'm not having confidence how to speak in front of you people please help me