Does the method chosen to carbonate your beer impact taste? For example, forced Co2 vs bottle vs natural, etc.
Not in my experience. I did a test where I used corn sugar, cane sugar, brown sugar, honey and DME (maybe even something else) and also force carbonated a split batch. After 2 months of conditioning, none of the tasters in a blind test could distinguish one from the other, and no one exhibited a preference for any one method.
The most common methods of carbonating your beer will not offer differing flavor profiles, but there are exceptions.
When force carbing, your only addition to the beer is the gas itself, CO2.
When using corn sugar, your addition to the beer is 100% fermentable, so the CO2 gas your looking for is created, with no sweetness (or flavor) left behind.
The exception: If you instead use an addition which is NOT 100% fermentable. A good example of this would be Honey, which comes in many varieties (i.e. Orange Blossom). Your Honey addition will not be entirely fermented, and there will be a mild residual sweetness and honey flavor profile left behind. Now, whether or not this impacts your beer's flavor profile as a whole, will entirely depend on the beer being made. You won't notice any real impact on your beer's flavor profile unless your Final Gravity is close to 1.010 (Assuming you have not made any adjunct boil/fermantation sugar additions).