I intend to relocate my washing machine and electric dryer within the basement. The dryer will then be about 2 feet from the furnace. Is that OK?
Check the specs on the devices. They will each have some amount of specified clearance, and you shouldn't get closer than that number (for the dryer it may be zero on the sides, but I bet the furnace will be at least a foot all around).
Beyond that, make sure there's enough room to get in between the dryer and furnace to work on the furnace (you don't want to have to start furnace work by dragging the dryer out of the way!).
Otherwise, you should be good to go.
As far as I know, as long as the electric and gas lines don't actually touch, there's no more reason to be concerned about the gas and electric being 2 feet apart as there is 20 feet.
If you have a gas leak from the furnace, the furnaces own ignition system (pilot light or electronic sparker) has as much chance of setting off a blast as the dryer.