If the federal government shuts down on Oct 1, only essential staff get salaries. Congress is considered essential.
After the government shutdown, there will be a blame game to determine who gets held responsible for the closing of national parks etc. Either or both sides could in principle suffer the negative hit from the shutdown. Each party has to do something to get the upper hand in the argument. The first to renounce their salaries during the shutdown will get the upper hand. They can say ” We share the pain of the shutdown with the American people and stand in solidarity with them. The other side is not suffering and is inflicting pain on all of us – they have nothing to lose and are advancing an extreme political agenda which is not in the interests of the American people.” The other side will soon capitulate and give up their salaries too. But the second-mover does not reap the rhetorical benefit. There is huge incentive to move first. So, there is a pre-emption game and everyone renounces their salaries.