A CAI does not cause the car to run rich unless something is wrong. A cold air intake causes the engine to pull colder, denser air (but a very, very very small effect overall), meaning that if anything, it would make the car run lean IF the ECU could not trim enough to keep the AFR in check. But that is not the case. The very slight difference in air density (again, a CAI is not doing that much, little to nothing on a tC) is easily made up for by the ECU adding more fuel to the mix. That is the whole reasoning behind how they work. Think about it this way, your car can run in temps way below zero, or over 100 degrees, the difference in air temp and density in those conditions is WAY larger than the difference in pulling air from the top of the fender level vs pulling it from a foot lower than that
If a few degrees difference in IAT that a CAI provides caused the AFR to be off even slightly, then your car would not operate when the temp dropped to 20 below zero. And running that smooth metal tube vs the stock in take does not allow the engine to pull in gobs more air like they like to tell you it does, or the CAI would make more than a measly few HP at the crank. In other words, the ECU and fuel system is more than capable with keeping up with the change in airflow or density.
If the car is running rich, the MAF is the very likely culprit. An oiled filter on a CAI can cause issues if the filter is over-oiled. But, the real issue with oiled filters is on an SRI, where the filter is closer to the MAF. The MAF does not get clogged really, since it is just a small sensor sticking down into the intake. The surface of the sensor gets dirty, which is what causes the problems.
1) Did you reset the ECU after you installed it? That may or may not help.
2)Remove the MAF, and clean with MAF cleaner. DO NOT touch the sensor element with anything, simply spray it out well with the cleaner and let it air dry.
3) Are you sure you have the correct CAI for the year model? A CAI for a 2005 will not work on a 2007+ for example, because of the MAF mount. You should have noticed that things did not mount up correctly if that were the case though.
There are vacuum lines you could have left off when you installed, but if that were the case, the car should be running lean since it would be pulling un-metered air into the intake.
For reference, what code exactly is the ECU throwing? Are you sure it is a rich code? And who is "they" ?
Not to be anal, but your fuel is not burning rich, your air to fuel ratio is rich. Rich means too much fuel for the amount of air, lean means too much air and not enough fuel.