What is the purpose of the roof drip moldings on my 2003 toyota tundra? Do I really need to install them?

They are cosmetic components to make the vehicle look better and can be done without if you don't mind the odd look of something missing.

I can answer my own question. You need the drip moldings on your vehicle. They are not cosmetic at all. As stated in my question, I just had my truck painted and the drip moldings on the roof were too brittle to put back on. I left them off and sure enough it happen to rain. I had water at both rear corners and water at the front corners. Actually, it was just wet or very damp. I put a wash rag at both rear inside corners of the cab. It was still getting too damp. I am trying clear packing tape on the front and rear area on the roof. Just a small enough piece to cover where the window seals and roof meet. So far, so good.

The missing drip mouldings are not the main cause of the leak. It is the glass sealants that keep out the water. If you install the mouldings without rectifying the leak at the glasses, leakage would still occur though the moulding would help to reduce leakage as water would not be able to accumulate with the moulding.