What happens if you take off with a direct tailwind?
Given how few bounding criteria you have in your question, the general result is


For fixed wing aircraft: your ground speed will be significantly
higher than if you'd taken off heading into the same wind, and thus
you'll tend to use up more runway before lifting off.

Example: with a take off speed of 100 kts, going into or downwind is the difference between 90 knots or 110 knots ground speed before reaching take off speed when the wind down the runway is 10 knots.  This difference becomes more pronounced as wind speed increases; it's a differential of 2 x wind speed.  

With a long enough runway it won't matter in terms of "taking off before running out of runway."  On a short field it may matter a great deal.    
For Rotary Wing aircraft: you may need a bit more power to hover when
lifting off with a tail wind.  The FAA has this guidance: 


  Headwinds are the most desirable as they contribute to the greatest
  increase in performance. {snip} When taking off into a headwind,
  effective trans lational lift is achieved earlier, resulting in more
  lift and a steeper climb angle. When taking off with a tailwind, more
  distance is required to accelerate through translational lift.



For a variety of aircraft types, if the tail wind is high enough your manual will recommend against/prohibit down wind takeoffs, but that varies enough between models and types that a general answer can't encompass all of the variations.