What determines the colour temperature of a bulb?
Is there any correlation between energy used and colour? Does colour determine energy consumption?


Minuscule, there are an awful lot of other design considerations that will have drastically larger impacts on consumption in both incandescent and LED.


  Is it possible to get a different colour temperature by putting a filter in front of the bulb? For instance is it possible to put a filter in front of a warm white bulb to get daylight?


Yes, but it is always going to remove intensity of the light at its destination. Take a warm light bulb, a filter (neodymium I believe) that filters out some of the reds and yellows and it'll appear more like daylight (bluer), but will be less intense when measured because you're blocking some of the source light.

The world is significantly different with the various materials used in LEDs - if you are interested in what makes the colors what they are, go to a white and then any color data sheet and it will talk about the different materials (AlInGaP, InGaN, etc) that are used to make it appear the color it is.

Note: this being a simple question, the answer has been simplified... there is so much more to the world of light!