Will a smog check place test my car if it has expired tags?

They could care less about your tags!

yes and a cop will likely impound it too. And give you a expensive ticket a expensive impound fee and a ticket for anything else he finds wrong. Go to the dmv they give you 60 days to get it done already. you can buy a trip permit or even short term registration to get smog. And its a lot smarter to pay a few extra bucks than a whole paycheck or even two paychecks. you need your insurance papers with the policy number for temporary tags as well. good luck do it sooner not later next time.
And do not buy cars from owners to cheap to register the car. They saved themselves some money and make it hard on the person buying the car. And sometimes there are other problems like unpaid tickets. And cheap dipsticks always skip oil changes and other maintenance. That is why I don’t buy unregistered vehicles.