I've managed to add a C wire to my thermostat. I then decided to check the voltage between the R and C wires, and find that it read as 27.8 volts, not the 24 volts it was supposed to be. Could this cause problems? I do have a Lux thermostat which is having trouble with it's ambient temperature reading, but I'm not sure if it's because of the voltage or something else.
It's a bit high but whether it will be a problem depends on your thermostat and how it tolerates the high voltage. Also consider that the furnaces controller board is also receiving this voltage.
Before you start anything else, check to see if your thermostat can be calibrated and complete this step first.
To rule out the higher voltage as the cause of the faulty temperature reading, find another transformer that gives you close to 24VAC and connect it to R and C - basically use that power source instead of the transformer in the furnace. If the temperature reading is correct now then you have verified the cause. If other transformers are also giving higher than expected voltages, check your input line voltage - if that is high, so will the transformer output.
If line voltage is OK and other transformers are OK then you can replace the transformer in your furnace; it's a pretty straight forward replacement usually. Just remember to turn off the power to your furnace first!
Normal voltage range for 24 volt HVAC systems is 20 to 30 volts. Normally if you are in that range to the upper/lower bound of accuracy of your measuring equipment you're fine. An induction transformer will read higher when lightly loaded. All of them I've ever worked with over the years do this, even modern switched mode supplies have a lower bound under which they read higher than the nominal voltage. As always check with the equipment vendor if there are any doubts regarding range of voltage over which your equipment is expected to operate.
Sounds like your transformer is set to 208vac, not 240vac, or if the transformer has multiple leads on it, the 208vac lead on the transformer is connected to the 240vac line, rather than the more correct 240vac lead. A 208/24vac transformer running at 240vac will put out just about 27.7vac (basically what you are seeing). 208vac is used when the hvac is running on a commercial 3phase electrical system, not residential single phase (240vac).