When you order a Lobster roll the Lobster Rolls are usually served cold and not hot or warm.
When Lobster Rolls are served in Maine they are served on New England-style hot dog buns which are split on the top instead of the side and have flat sides for grilling.
The meat of the lobster roll is served cold and mayonnaise is typically spread inside the bun or tossed with the meat.
So when you make lobster rolls at home you can serve the lobster rolls cold if you want too but if you prefer to have the lobster rolls hot or warm when eaten you can do so as well.
It doesn't really matter but I prefer the lobster meat to be hot and not cold because I find the lobster rolls more enjoyable when the lobster meat is hot and not cold.