Measuring and Replicating Electrical Codes from heating controller
Replicating the signals will depend entirely on the method used to encode them so it will make sense to concentrate on that first. As Jim Dearden suggested in a comment it would be worth asking the manufacturer, they might be forthcoming or they might treat it as proprietary and confidential.

Failing that the usual piece of equipment to use would be an oscilloscope and the best choice would be a digital sampling oscilloscope with a reasonable deep capture length so you could capture and record a decent sample of the signals for viewing. 

Dedicated / high quality scopes can be expensive but on E-bay a search for "USB oscilloscope" you'll see some ranging from $25 upwards. Some of the cheaper ones only offer up to 3KHz bandwidth which may or may not be fast enough depending on how the signal is encoded, while the data rate is probably low it might be modulated onto a higher speed carrier to get power and data over the same cable. I'd suspect though any of the units that can measure in the MHz range would be more than sufficient.

Another "almost free" option would be to try using the soundcard on your PC as an oscilloscope. I'd probably only recommend doing this using an old laptop or USB soundcard in the event things "end badly". There are several projects for this floating around on the Internet and they might or might not work well enough for this purpose. Here's an example I found:

http://www.ledametrix.com/oscope/

You'd only need to build one channel of that using a pair of resistors for what you're doing. I'd recommend though for a start changing the 82K resistor to a 1M odd because sometimes "low voltage" is used to refer to anything below 50V. If you can see a reading but it's very small maybe try halving the resistor a few times to get a decent range. The two resistors are just forming a voltage divider to get the signal input a reasonable range for the line input on a sound card. There's a simple voltage divider calculator here so you can see how it works:

http://www.electronics2000.co.uk/calc/potential-divider-calculator.php

The sound card may not have a fast enough sampling rate to capture the signal but it may be worth trying for a start. One additional disadvantage to using a soundcard is that you won't know the actual voltage of the signal accurately, but you could for example get a fairly good idea by comparing the readings against what you get from say a fresh 1.5V battery.