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.
Maybe you are plan B, but I think that setting a deadline for them is more likely to backfire than help. The easy thing for them to do is to withdraw your name from consideration - especially if you aren't the top candidate.
Don't give up other opportunities and make your own decisions on what you think best for your career, but I urge patience here. They may have entirely different reasons for delay.
You can't and shouldn't try to "set a deadline". Instead, just let them know gently..."I am getting other offers and need to decide soon". Also, "I am very interested in this position, but may need to accept another one, if your offer is delayed too long". Given the market imbalance, I don't think competing INTERVIEWS are important or compelling. However, other OFFERS obviously are. And you are being responsible to let them know you may become off the market to them.
Realistically, you're probably not getting contacted because you are not high on their list.
But there is some chance they are just slow or bureaucratic. So, yeah...reach out gently to inform/check with them. But it should be more in the mode of letting them know you are in demand. Not "setting a deadline". Of course, if you have an exploding offer from someone else, let them know. But again, you are not "setting a deadline". Just letting them know when the great candidate might become unavailable.
My experience in hiring/getting hired is that this is generally positive...to let people know that you are in demand. "Speculation drives the market." But be gentle and smooth about it.