Some people have suggested that I remove my cars catalytic converter. This is because my car is designed for a different market, and the manufacturer recommended fuel is not sold in my area. It is common for cars here to have the cat get clogged and the only options are to either install a new one, which is expensive, or to completely remove the cat, which is not a legal requirement here. Are there any complications which could arise due to removing the cat, apart from the composition of the exhaust fumes?
tl dr: There would be no issues with the running of the vehicle.
If the car in question is equipped with OBDII (or the international like), where the cat is being monitored by a second lambda sensor (O2 sensor), the only side effect of removing the cat would be the check engine light (CEL) would illuminate to show the secondary lambda issue. This would be more of an annoyance than a problem, but it will remain lit all the time. This would preclude you from seeing it light if a real problem occurred. The only way around this would be to put in a replacement for the lambda sensor which would fool the computer into thinking everything was okay (there is a term for these fake sensors, but the name eludes me at present), or to have an aftermarket tune put into the computer, which would set the thresholds for "out of range" on the lambda sensor so the CEL would never illuminate.
The other answers have addressed the environmental and health concerns, however, I want to add a few points.
Firstly, if the specified fuel is not used, this can cause more problems than just a damaged catalytic converter. You should address this as a separate issue. Most manufacturers will specify a minimum requirement and recommended standard - don't confuse the two. Read about the different rating standards here: Octane Rating wiki.
Secondly, Find out whether your fuel stations use RON, MON or AKI ratings. If you are unable to use the recommended spec but can use the minimum spec, then there should be no harm to your catalytic converter.
This assumes that your fuel suppliers and fuel stations adhere to these standards. I know some parts of the world don't have good standard enforcement and fuel can be mixed with water or other contaminants. This is a separate issue, so please update the question if needed.
Thirdly, removing the catalytic converter may trigger a fault code which will illuminate the check engine light (assuming you have one). You can get around this by fitting a spacer on the downstream lambda (O2) sensor. This way you will have a normally functioning check engine light should you have any actual faults.
Finally, I would leave the catalytic converter in place until you are certain it is damaged. I would only remove it if it becomes clogged/blocked causing performance issues. In that situation, replace it with a free flowing silencer box / resonator / muffler. If you replace it with a straight pipe it may make the engine sound become undesirable. Make sure this is done properly with proper flanges and gaskets to prevent the exhaust blowing.
Is it a good idea?
To address the literal question in the title, no. Just because one can doesn't mean one should.
Not for the environment
The cats are there to scrub out the most of the CO and NOx nastiness that is the byproduct of internal combustion. If I remember well, these chemicals contribute to acid rain.
Not for your health
There's a pretty good chance the fumes will find their way into the vehicle cabin, especially if your windows are rolled down and/or you have an exhaust leak. These fumes don't just smell bad - they can induce headaches as well.
Removing the catalytic converter will save a lot more money than replacing it, and will make your engine exhale better and last longer due to the removal of a suffocation device "cat". Only issue outta all the ones I've removed it is if the downstream O2 sensor is plugged directly in it, which there are ways around. If that is the case then the first option is, remove the "cat", gut it out completely, reinstall it, and plug the sensor back in. Option two: remove the "cat", fill in that gap with exhaust tubing, bore a hole in the new pipe that the O2 sensor will fit snugly in, and there should be no engine light. As long as that sensor has hot gas exhaust flowing over it, it's not gonna trigger an error code to the car's computer. If there's no sensor in the cat, just chop it off and fill in the gap. It's gonna run a lot better, I'll tell you that. I do it to every car I've ever had and ever will have.