Not en masse. If researchers refused to review en masse, the system would break down and no papers would be published (or they'd be published without review) and the impact would be very visible.
On smaller scales there has been a Cost of Knowledge boycott of Elsevier, an individual publishing house. This was initiated by Timothy Gowers in 2012, in protest of Elsevier's perceived high prices and gross margins. The target is Elsevier because it is the "worst offender" - Elsevier's margins are very high, above 20%. The boycott allows the signatories to say if they're not willing to publish with, to do editorial work, or to referee for Elsevier. It hasn't been effective. AsIt's had minimal impact: as of time of writing, there are 16,946 signatories, a tiny fraction of the number of academics in the world (millions). It did, however, lead to Elsevier lowering its prices slightly for mathematics journals as well as making some old papers available for free.
On an even smaller scale, I've seen researchers decline to review papers because they don't review for non-OA journals, or because they don't work for free. This is a very small fraction however: I've invited hundreds of reviewers, and the number who decline for these reasons can be counted on one hand.