How can I stop my print bed tape from sticking to the filament?
This can highly depend on the slicer you are using. Some software such as Makerware and Slic3r allow you to adjust the settings for the first raft/part layers. I might suggest adjusting this "Z0" point to about 1/4-1/2 of your layer height. Essentially the first layer (or two) will not adhere as well.

This is just one suggestion of many solutions. Here are some other variables I could think of off-hand:


Type of build plate tape (ie masking, painters, kapton, etc.)
Type of material. I've noticed that PLA is very stubborn if you let the part completely cool after printing and that it's much easier to remove the part from the build plate/raft right after it's complete.
Type of build plate. Are you applying too much heat (if you have a heated bp) for the material such as PLA?
Try lowering your layer height. This will ensure that each strand does not have too much surface area and therefore less chance that it will create a vacuum affect with the build plate. This can, however, result in a worse surface finish.