If I was reviewing a paper, and the authors failed to cite important literature, I'd recommend rejection until the authors provided correct citations. It's the authors' responsibility to provide appropriate references.

I assume that many other reviewers would feel the same.

I wonder how reviewers would react if authors of a submitted paper refuse to cite paywalled papers. Is it a valid reason to refuse citing some papers?

Certainly not. There are a few areas where this could be OK; for example, if you are citing expository material for background (rather than to assign credit), then you can choose whichever sources you feel are best.

However, in many cases you have a scholarly obligation to cite papers, for example to give credit to people whose work you are building on, and there are no acceptable grounds for refusing to do so. It's a serious form of academic misconduct, even if it is done for idealistic reasons.

If I ran across an author who refused to make necessary citations, I would be extremely displeased, and I would recommend that the paper not be published until the citations were included. I would not fully trust that person's judgment in the future, and I would be suspicious that other papers might be missing important citations.

Instead of omitting citations, you could add some brief commentary about the lack of open access. (Reviewers or readers might dislike it, but it's in no way academic misconduct.) You should be very careful with that, since you could really offend an author who has made the paper available, just not where you looked. For example, it might be in an institutional repository. If you want to avoid giving offense but still encourage open access and help readers, you could give suitable arXiv or repository links to each paper for which you can find them.

If you would like to boycott paywalled journals, go right ahead - that is your right. As a reviewer I would reject your paper for not citing relevant sources. As a reader of your paper and/or author of a paper you didn't cite, I would be severely antagonized as well.

The bottom line is, your ideological battles should not be waged on the backs of honest readers and authors who dedicate their lives to producing and disseminating good science. But if you want to commit career suicide, be my guest - no one will stop you.

Edit: the saying "Be the change that you wish to see in the world" also comes to mind. In that vein, if you don't like paywalled journals, the honorable course of action would be to simply not publish in them yourself. This would be vastly superior from a moral, ethical, and philosophical point of view to waging some kind of take-no-prisoners, collateral-damage-be-damned nuclear warfare against them, which is effectively what your question is proposing.