Carry-on luggage is usually not weighed, there are simply safety margins in place for that sort of thing. (A human probably isn't capable of carrying a bag heavy enough to erode those margins.)

And fuel burn is the least of the concerns when it comes to weight - a far bigger issue is how it affects balance and engine lift. But on a plane the size of a jet airliner, this too is almost never a real concern when it comes to passenger weight - a standard weight is just assumed for each passenger and their carryon. (It's a concern for cargo weight, but that's why cargo is always weighed and balanced properly.)

Maybe if the airline notices that all the passengers are sitting on the left (for some strange reason) or something like that, they would reseat people.

Cant recall ever being weighed as a passenger and in this day and age you have passengers who are 120 lbs and 300 lbs yet you get penalized if your suitcase is 1 lbs over the limit.

Running out of fuel isn't the real concern. It's more about center of balance and aircraft performance (ability to climb with an engine out).

Wouldn't it be possible to install strain gauges in the landing gears to determine the plane's weight?