What is this and can dryer work without it?
It looks like the moisture sensor. It detects damp clothes and keeps the dryer running.  They are normally attached on the back of the drum. I have replaced the control boards in the past that have failed but never had the sensor come out. I know the dryers I have repaired worked on timer but not auto. I would look to see if it snaps in place with spade connections, if the electrical connections are fixed in place it would probably be ok to use but if there are any loose / flopping wires I would get it fixed prior to using it or other electrical problems may be caused.