"How can I test that without taking the dash out?"
I don't know where the inside air intake is or where the door is, but they are usually under the middle of the dash or over the passenger footwell on the lateral side. (There is just too much other stuff on the driver side, like the steering column, brake and accel pedals,....) Often the dash has to come out, but you should be able to tell if it is opening by looking and listening.
Does the blower sound different when you change from outside to inside? Is it very cold where you live? If so, the temp of the air should change when you switch after the interior is nice an toasty. Stick you head under the dash and turn the recirc control and listen. You might be able to hear or see the door opening and closing. It is letting interior air in after all.