It can be sad to get rid of a shrub, but sometimes they have to make way for new landscaping plans. Aside from the tedium of digging it up manually or giving it away (requiring the recipient to do the digging), there are other natural, environmentally friendly ways of getting rid of an unwanted shrub.
Prune the shrub as close to the ground as possible. After using hedge trimmers, you may have enough access to the stump to use an axe or saw.
Hammer a copper spike vertically into the centre of the trunk. Such spikes are normally used to hammer down copper roofing and can be purchased at a hardware store. Autumn and winter are the best times to do this, because the shrub’s nutrients (and the copper) will then be pulled into the root system. The copper spike can help disintegrate the roots, which is important to avoid a revival of new shrub shoots the following spring.
Cover the stump with an opaque tarp to deprive it of sunlight after you've driven the spike in.
Pour bleach into the stump around the spike and the roots. Do this over several days. Alternatively, pour salt into the open stump. This is not as good for the soil as the above steps, and it is certainly optional. However, the more approaches you try, the better.
The fastest, most efficient way to remove the shrub and ensure that the roots are removed is to use a backhoe, if you can afford the rental expense and don't have the time to try the above steps. If you choose this option, make sure the heavy equipment won’t damage the rest of the landscape.
Do not bother burning (or "top-killing") the shrub. The roots will remain and the shrub will grow back.