Should I remove the SD card and battery from my DSLR after each use?

The only reason for removing a rechargeable Li-Ion battery would be to prevent it from being totally drained. A totally empty Li-Ion battery can not be recharged by most chargers. This is due to the increased risk of fire when applying voltage to such batteries. In most cases, it should take weeks for a battery to be drained that low. So there is no need to remove the battery each time you are done with the camera. If you store the camera for extended periods then it is a good idea to remove the battery and charge it periodically to prevent it from going completely dead.

If your camera uses a different type of battery than the typical Li-Ion type found in most current cameras, then you would need to consider the battery type.

• Alkaline and older style Carbon ("Heavy Duty") batteries should be removed if you're not going to use the camera for more than a week or so. Leaving them in for long periods can lead to corrosion leaking from the batteries and getting into your camera. Leaving them in also discharges them slowly as the camera draws minimal amount of power to maintain the time/date clock and other information stored in volatile memory.
• Older rechargeable batteries, such as NiCd and NiMH batteries, are similar to alkaline batteries. Long term storage can lead to total discharge and leaking of corrosive materials.

On the other hand, there are reasons for leaving batteries in the camera when it will not be left unused for extended periods. Most cameras have a small capacitor on the main board that powers the clock while batteries are being swapped. These capacitors are charged when a battery is installed in the camera and discharge to power the internal clock when a battery is not present. After several days, the capacitor will be exhausted and the time and date will need to be reset when a battery is next installed in the camera. Other information, such as the next available file number or current folder number may also be lost when the internal capacitor can no longer power the internal volatile memory.

As far as SD or other types of memory cards are concerned, there's no need to remove them when storing the camera.