Can I remove the fresh air supply that is attached next to my furnace?
All fuel burning devices need an air supply.  In the case of direct vent appliances, there is an air intake to be plumbed to the outdoors as well as an exhaust pipe.  This assures that the appliance is not drawing air into the house through windows, doors, etc. defeating weatherization and sealing.  The combustion chamber in DV appliances is closed and isolated from the room.  Even a direct vent fireplace is completely enclosed to allow no air to enter from&mdash;or exit to&mdash;the room.

Direct venting is still largely the exception rather than the rule, though DV is gaining favor with many people who prefer higher energy and cost efficiency, so much new construction is now using DV.

A 1950s gas water heater has an air intake, but it simply draws air from its surroundings.

That water heater does not appear to be direct vent, so a general air supply for it is needed and the "floating beanstalk" air tube is a reasonable way to achieve ventilation&mdash;removing it would be "bad".

It occurs to me to try to decrease the chilling effect the intake causes for the room by moving it closer to the water heater, and maybe even "boxing it off" somewhat.  However, if the outdoor air is very cold, like below -30&deg;C/-22&deg;F, and there is a lot of airflow, such an arrangement could damage the plumbing and tank by freezing water in the pipes, etc.