The federal government has a $120 billion plan that would cut Amtrak’s Acela Express travel time between New York and Boston by 45 minutes, making it a two-hour-and-45-minute trip.

To get it done, however, it is going to first have to get through Old Lyme, Conn., where opposition is strong to the project that would bring four new track lines and as many as 110 trains a day under its historic downtown in a tunnel.

“That...