For an organism (think single-celled) living in just the right kind of environment, it might just be possible to survive using only facilitated diffusion, at least as far as small molecules are concerned. This kind of organism would have to maintain exactly the right concentration of the molecules it wants to keep or get rid of, based on the external concentration of those molecules.
For example, if an organism was living in a glucose-rich environment, it might be able to fulfill its glucose requirements by (1) making the membrane itself tightly impermeable to glucose and (2) tightly controlling the number of passive glucose transporters.
It would be nigh-impossible to achieve this for every solute in the environment, especially those that are able to permeate the membrane.
If there were such a thing, it might be found in organisms that live in very low energy aquatic regimes such as anaerobic methane oxidizers, where the solute concentration in the environment doesn't really change in small time scales.
Even in such environments, the number of small ions in the environment is likely to be high enough that some active transport would be required to maintain the osmotic balance within the cell.