I just moved into a new home, and the flower bed drip irrigation is buried under mulch and about 2 inches of sandy loam. This isn't a correct install is it?
There are a variety of drip irrigation configurations, some are meant to be underground while others are meant to be above ground.
An example of underground drip system could be found at: http://www.dripworks.com/category/dripworks-drip-tape
This type of drip tape could be buried anywhere from 0" - 5" of depth.
An above-ground system would be an irrigation system that you could get at most hardware stores.
The mainline of above-ground drip systems could be buried while the emitter lines and drip emitters should be above-ground.
There is no correct way. Some dripline has special emitters that can't be clogged by roots. It might be great for some plants, but would you really want to bury this and let it become a permanent part of a tree's root system?
For transport runs it's a trade-off. Ease of maintenance or modification. Root entanglement. Tool damage. Sun damage. Rodents. Etc.
I handle my yard/plant/equipment tradeoffs as follows: Rig a surface system first. Offset long tubing runs to the side by the depth I plan to bury them later, to create the needed slack. I don't bury sections with emitters or barbed inserts, just transport runs. Once it's all set up and tested I go about selectively burying the annoying runs across the yard. Depth depends on where, how long, how likely I am to modify, and whether it will entangle tree roots.