Lobster rolls should be refrigerated as soon as possible when you're done serving them or eating them.
The lobster rolls can only sit out at room temperature for no more than 2 hours before it goes bad.
Once you cook any lobster including making the lobster rolls you should refrigerate any leftovers of the lobster rolls or lobster within 30 minutes to keep the lobster from going bad.
But as long as you refrigerate the lobster rolls within 2 hours the lobster rolls will remain safe to eat but after 2 hours the lobster rolls can start to grow bacteria rapidly which can make you sick.
So refrigerate any lobster rolls that are leftover soon after you cook them and finish eating and the same goes for any other meat.