Typically what you're asking for is non-trivial. There are some expensive middleware packages for this feature, but not a lot of free ones.
A straight-forward (but still non-trivial) approach is generate pre-broken versions of geometry and then to switch to those. This requires more art investment and has a number of limits, of course. Good investment in tools can improve the workflow but will take some time to get up and running.
Difficulties of this in general are those objects are 3D (with multiple "materials" internally). If you blow up a hill, you don't just want to depress the grass on it. You want to generate dirt, stone, and clumps of turf that landed nearby. To put a hole in a wall, you need to know what the inside of the wall looks like (walls are not infinitely thin planes in real life). Putting a hole through an entire building in your scenery is that much harder. Handling physics appropriately (e.g., ensuring a structure collapses if you remove its supports) is even harder still, since you need to model the structure intricately both visually and physically.
You can hack around a lot of this with some basic assumptions and tricks. Walls can have "decal" objects that are placed around a hole to make it look rougher and hide/cover the spaces between the inside and outside of the wall. Generating the hole in real-time is still a trick, though; AAA solutions precompute these holes, often with high-powered algorithms and very extensive and complex tools integration.
The difficulty of properly handling deformable terrain is partly why you almost never see it outside of a few precomputed cases and games like Scorched Earth (which is all 2D). It's possible, but not easy, and potentially unfeasible for complex types of terrains and structures.
Note that both UDK and Unity use NVIDIA's PhysX, which is one such middleware with "destruction" capabilities. UDK includes a "fracture" tool in UnrealEd using the APEX Destruction toolkit with PhysX to do things the precomputed way mentioned above. It's up to you to decide if this is "good enough" for your needs or not.