If a bed is placed over a heating vent in the room farthest from the furnas, could the heated air be enough to cause a fire? This is in a mobile home that is a 14X76.

I would say probably not. If the temperature of the furnace were enough to ignite that material, then it would do so around the entire mobile home. Also, there would be warnings not to put anything over the vent like that near where the vent is.
I'd think with paper needing 451* Fahrenheit to spontaniously combust,(Thank you Ray Bradbury), other materials would require a higher range before they too would burst into flame. And if the heat of the furnace were near that temperature, you'd certainly be very uncomfortable.
Try this. If you have any type of thermometer, like an outdoor one, bring it inside and put it near the vent. If the reading maxes out, then move the offending item. If not, then there is no worry. Or, put some paper near it and see if it gets to 451*.
I think that you will do fine, and maybe even enjoy the fact that the vent is so close to the bed. Just keep it clean as possible, so that the smell of cooking dust bunnies doesn't bother you.
I refuse to brag that it reached 81* here today, and the internet weather thing says it is currently 72* outside.
But it did.:p

It won't cause the bed to ignite but it could cause some temperature issues in that room. Generally it is advisable NOT to obstruct supply or return registers.

They make adapters for floor vents that sit on top of the register and direct the air out in a different direction. They are very low profile and are used for directing the ait away from furniture. Extends to 37 inches long see below.


SkyMall - Floor Register Extension