Can you add furring to 2 x 4 walls to hold 2 by 6 insulation?

You can for sure add 2 inches to your 2x4s so they can accommodate 5.5" batts. You would need to bring out your wiring boxes to mee the new thickness and it would inset your windows more, but there is no reason we can think of as to why that would be a problem otherwise, assuming the wall is currently solid.

A few questions though - where is it and how old is it? And is this a house or an old cottage? I don't know if you experience heavy snow loads where you are, so my only concern without knowing some variables would be making sure this building can continue to carry the load. Weather patterns are far less predictable now, and there are more frequent and heavier precipitation events, so before you do a bunch of work to it just be sure it's ready to last into the future. If you do recieve a lot of snow and it is a 2x4 wall on 24" centers for example, you may want to take advantage of that added 2 inches of wood to help carry a bit of the load if necessary. But, all this is simply a long-winded way to say, yeah, you can add 2 inch furring to a 2x4 wall and get more insulation in there. 

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