Only in the respect that it can use up your data at a faster rate.
If for example you ask an app to give you the latest information on weather or news, that information gets delivered to your phone faster but should not take up any more data than using 3G or 2G, all that should change is how fast the data gets delivered and the cost Mb for Mb should be the same.
That said when I’ve done speed tests using 4G then 3G on the same network the 4G speed test most of the time uses more data but under normal everyday use all thing should be equal except the speed of delivery.
Why most folk think that it’s more expensive is because all of a sudden changing from 3G to 4G your phone can be used to watch SD or even HD video whereas previously it struggled using 3G so you did not bother.
So bottom line from my point of view is that the cost should be the same there and about but you will probably use it more so will be more expensive than when you used a phone on 3G.
That’s my thoughts.