Can open-source software be peer-reviewed and published? My colleague and I have developed a software tool and intend to release it open-source.

This tool is specifically for tasks in my field but we think it would be helpful for the wider community. Our institution will permit us to release it provided we get appropriate credit.

Thus, we wish to publish it in peer-review. Is peer-review publication of domain-specific software available? If so, what is required to publish it?

In this case we intend to publish the method and tool on it's own merits without supporting data or an application.
Yes, Open Source software can be published. What's required varies depending on the venue.

There are general journals that focus on the software process. The idea is to encourage better software development gets the credit it deserves. Examples of journals with this approach are the Journal of Open Research Software and the Journal of Open Source Software.

Then there are domain specific journals that have specific software paper policies. An example would be the AAS Journals which state


  AAS Journals welcome articles which describe the design and function of software of relevance to research in astronomy and astrophysics. Such articles should contain a description of the software, its novel features and its intended use. Such articles need not include research results produced using the software, although including examples of applications can be helpful.


(Emphasis mine to link to a point in the original question)