Whenever I work on project where I get to choose colors and color schemes, I get really excited. I think color theory and meanings that get associated with colors are super cool!
Personally, I always refer to something like the image below for starters. Over time, you start to memorize what color is associated with what and you no longer need the chart, but I still like to look at them anyways.
Need: Red is often used to create a sense of urgency. It can sometimes generate negative connotations though.
Should Do: Perhaps Yellow (often associated with caution) because users should pause and consider if they want to do this task.
Want: you could go with Orange here (ambition), but I think Green also fits (envy). Or you could go with a pink (love) which is also like a "lesser" version of the Need to do Red.
I'm thinking, if you can come up with a way to rank the items (by Need/Should Do/Want and then an order within those) you could use a gradient color scheme. (Maybe red fading into yellow, or blue fading to light blue, etc.) The most important (or #1) task to do is the most saturated and appears at the top. You won't get three distinct colors for each category, but it will alieviate the cohesiveness issue with having three colors.
You may want to consider having a default with the colors your believe associate best with the meanings you're going for, but leave the user the option to set their own color scheme. Unfortunatley color meanings aren't exactly concrete since it's largely based on perception which varies across people, regions, and cultures.
You can get a general idea from charts like the one I posted here, but in the end everyone has their own personal preferences and works a little differently. Personally, I come up with new color schemes for my own task lists all the time and only I know what the code is because I never write it down. (sometimes it's the number of stars next to an item, sometimes it's the color of the ink, sometimes I highlight, etc.)
I would love to see what you come up with when you're done :)