What happens when an FPGA is powered on and left unconfigured?
This is an interesting question. From personal experience, I have left FPGAs powered up for hours in an unprogrammed state while performing checks on the rest of the circuitry when a new board comes in from the assembly house. I have not noticed any detrimental effects from doing so. But I've honestly never thought about it.

I looked around to try and find a recommendation from an FPGA manufacturer but could not find one. The only statement I found regarding this state is from a Lattice whitepaper and regards the design of the FPGA itself and not how it should be used:


  Pre-Programmed Static Quiescent Device Power Consumption is the amount of power 
  consumed by the FPGA prior to the device being programmed.  For quiescent device 
  power consumption the FPGA is in a non-programmed state, yet has been powered.  It 
  is important that the device not consume significant power during this time, as 
  conceptually the FPGA device could draw excessive power and potentially shut down 
  power supplies, preventing the board from successfully initializing itself and the system.
  The FPGA supplier therefore has to carefully design transistors that have a low static 3  Designing for Low Power 
  A Lattice Semiconductor White Paper 
  power footprint, without compromising in areas where higher performance is required 
  (e.g. I/O and SERDES)


Xilinx likewise mentions what the quiescent current is so that you can design your power supply accordingly. But does not mention what the effect on the device is of leaving it in such a state:


  Static or
  quiescent power is mainly dominated by
  transistor leakage current. When this current
  is listed in data sheets, it is listed as ICCINTQ
  and is the current drawn through the
  VCCINT supply powering the FPGA core.


I would be very interested to hear if anyone has experienced damage to a device from leaving it in a quiescent state. But I think as long as the power supply is properly matched to the device, there should be no problem.