I didn't hear back after a postdoc interview. Can I set a deadline for a response?
You should let them know that you have other opportunities at other universities that you need to respond to, and what those deadlines are.
If they can, they will try to avoid the situation where they can not hire you because you have been forced to agree to another job. That does not mean they can.
Generally speaking, people being informed is good.  Setting deadlines that look arbitrary is not.