The policies of any journal should be public and consistently applied. Having a policy that you have to be a reviewer before you can be reviewed is irrational, I think. There are many rational reasons to refuse any particular paper for review. But if you have a policy that the complete refusal to review for any reason, including that you want to be paid, could lead to desk rejection is a different matter. 

One large problem here is how to state such a policy so that it isn't too narrow. A policy that complete or consistent refusal is not acceptable is difficult to state specifically. It needs, in my view, to be stated as an expectation. _We expect that our authors are available to review the works of others._ That is (approximately) what you want, but applying it too rigidly is a mistake. 

But then, the question arises about how to handle such situations in practice. 

Actually, the journal should handle a problem like this much earlier than the situation described. It may be that the "reviewer" just doesn't know how the game is played and that reviewing is seen as a contribution to one's peers. To be a professional is to offer this service and, by cooperating, others will, hopefully, do the same. 

Therefore, send a letter to the person when they first refuse for this reason, explaining the process and the fact that reviewers are never paid and that paying them would increase the costs to readers as well, given that there are more reviews done than articles published. 

At that time, let them know that, while it is possible to refuse any given paper for review, if they refuse to participate in the process at all, then their future work won't be accepted as a matter of policy. 

I have refused to review papers for a variety of reasons, often because I felt I didn't know enough of the subject to do it justice. No one should have to make reviewing their first priority. But if they refuse to participate in a society as constituted they won't be welcome in it. 

They are, of course, welcome to work for a more rational system if they like. The current system has many practices that should be addressed, but we have, so far, presented only partial solutions, and most have their own negative consequences.