Can I put a postdoc as one of the references in my CV? I'm applying for a postdoc (Computer Science) and some positions ask for the contact information of 3 references. Beside my Ph.D. supervisor, I work with other departments in my institute, but mostly with the postdocs who were in charge of the projects. For instance, the professor head of department X knows who I am and that I did/do collaborations with them, but has no idea about the details. However, his postdoc researcher can provide much better detail regarding our collaborations and etc.

Nevertheless, I have a feeling that the employers (especially in academia) rely on professor-level references than a postdoc. So, I'm not sure about this situation.
This depends on what is available to you. Yes, institutions value the opinion of well established researchers (professors), but they also value the honest opinions of people who can speak about your likelihood of success and how you "get along" in the research environment. The best are people who can do both, of course. 

But you don't want lukewarm praise from top people. So think about who you have as possible recommenders and what they might say. Then just make a decision. I doubt that it will matter in a critical sense, actually.