The answer given by Ed is correct in saying that you can use a nailer, but you will have to use it more slowly. It won't recycle as fast as the manufacturer intended. Also a nailer actually needs two bursts of air to work - one to drive the nail and a second one to pull the plunger back into place. If your compressor is so undersized that it can't even supply that much air at a high enough pressure, your nailer may frequently jam. Of course I think that would have to be an exceptionally small compressor being used with a framing nailer.
High SCFM Tools
It will largely depend on the tool, but in general it will work slower or not have as much power. For instance, a friend of mine had a very high performance die grinder he used in the aircraft industry. I used it on a small compressor, and it worked, but not close to its rated speed, and then only in short bursts. I made the cuts I needed, but it took way longer than it should have.
Something like a paint spray gun might fail to work properly at all with the wrong amount of air. Other mechanical tools might also be so under powered that they basically don't work, but that's going to be more rare.
So, in short, if you can borrow a high SCFM tool and see if it "works for you" then have at it, but I wouldn't spend money on a tool knowing it would be crippled at best. There are electric alternatives to most air tools if you don't want to to invest in a large compressor or don't have the room.
With all of that said about how a tool would (under) perform, you also have to consider overworking the compressor. Many smaller compressors might be designed for a 50% duty cycle (easy math example, check documentation) which means the motor runs to build pressure in the tank for 45 seconds, and isn't expected to come back on for another 45 seconds. If you're using a higher-than-allowed SCFM tool, the compressor could run almost constantly. This might be OK for a few minutes, but will certainly shorten the life of the compressor if done in excess. Larger compressors have higher SCFM ratings and a higher duty cycle in many cases.