Historically 4x4s had a bad rep for their fuel economy but they aren't necessarily the gas guzzlers they used to be. These days, models like the Range Rover Evoque and Mercedes-Benz GLC-Class achieve a decent 33-67 combined miles per gallon (mpg) for city driving.
You're also likely to find that they're typically in a higher insurance group. Generally speaking, the higher the insurance group, the more expensive it is to insure. This is due to the likelihood of replacement parts being more costly, the bigger engine size, and their size meaning they could possibly do more damage when involved in accidents.
With these bigger engines, you might also find they have more expensive MOT and servicing costs. It's worth looking into this before you buy one.
4x4s can still pump out quite a bit of CO2. Check sites like Parkers to see what tax band your car falls into and take this into consideration for the overall costs. This makes shopping around for the cheapest 4x4 insurance all the more important.