I lost my fuel cap a couple months ago and haven't bothered finding a replacement. I haven't noticed any issues with it, but I'm wondering whether it's causing any damage or reducing gas mileage or anything.
It might cause issues with the evaporative emissions controls as the tank is currently permanently vented to the atmosphere and that can trigger a check engine light. In general it shouldn't be affecting the fuel mileage, though.
The other concern is that you'll probably end up clogging the various fuel filters sooner because the lack of fuel cap means all sorts of dust might enter the tank while you're driving. You'll also see more fuel evaporate over time, especially in hot climates.
Of course it also makes it a lot easier to stick a hose down the filler neck and siphon some fuel out for those who like to circumvent the legal process of purchasing motive fluids.
In the UK, most petrol (gas) stations sell temporary fuel caps in the appropriate colour, for very little money. They are universal fitting and usually made of plastic, so just press into the hole making a snug fit.
In addition to what Tomo suggests though, I would say that it also depends on the vehicle and the fuel that you use.
A big problem for motorcyclists is spilled diesel from vans and lorries. On many lorries there is very little height between the high fill level of the tank and the filler cap, so if the cap is not refitted correctly, diesel can slosh out of the tank and onto the road, causing a major hazard for other vehicles, especially two-wheelers.
Petrol leaks do not pose the same issue for other road users, but even so, if the filler cap has a short 'neck', then you could lose more fuel from it sloshing out than from evaporation.
Also, don't discount the threat of fuel theft. With many vehicles, it is the filler cap itself which locks the petrol tank, so going without a cap could make it trivial for people to siphon off your fuel. If the missing cap is hidden behind a flap the threat is less, but even so if someone is intent on stealing fuel, they will just flip open the flaps that aren't locked.
Very on most cars and extremely on some.
Extremely: on my car (2004 Subaru WRX), a loose filler cap will trigger a check engine light and the limp home mode. The engine computer will drop into a mode where it won't allow revs above a low limit and disallows all boost. Suddenly, you're in a 8:1 compression ratio car and aren't able to get out of your own way on the highway. This isn't a gradual switch, either. All of a sudden, you feel like you're going backwards through rush hour traffic....
Very: the fundamental purpose of the modern fuel cap isn't to keep gas from sloshing out or to keep sudden rainstorms from getting in. It's to keep the evaporating fuel from wafting out of the tank, wasting your money and leaving a trail of raw pollution in a cloud behind you. Remember, you're driving a car that gets progressively hotter the farther you go and gasoline is happy to completely evaporate as its temperature increases. Modern caps click on because they're trying to make a gas-tight seal against some non-trivial vapor pressures.
Get a new cap. They're cheap and, if you're concerned about siphoners, you can get one with the little luggage key lock. It won't stop a determined fuel thief but it will persuade a random punk to move onto the next car, who happens to be driving around with an old rag stuck down the filler neck as a temporary cap....