I suppose it depends on what they are being asked to "shut up" about. Leave voters have to accept that, unless circumstances change drastically, the result on 23rd June is going to be honoured, and they have at least to "shut up" about trying to stop the UK from leaving the EU as a whole. Scottish people are welcome to not shut up about it if this leads to a further Independence referendum but triggering of Article 50 will have to happen now. By contrast the fate of UKIP in 2015 could in principle be undone only five years later if supporters turn out in further numbers in 2020.
Complaining about the result, or the legitimacy of it, is very different from worrying about the outcome and trying to shape it. The concerns of Remainers surely have to be addressed as far as possible. In practice, this would presumably mean negotiating a deal that demonstrates that Project Fear was, after all, total baloney. Which would be nice.
But anyway. It's not a democracy if the only time you are allowed to be heard is at the ballot box.