Yes, the answer is always yes you need a DR plan if you are running mission critical systems. The level and detail of the plan just varies with the need.
The simplest DR plan for AWS would be to back up your data and images in reliable ways. For example, put the data in S3 so one region being out wouldn't stop you from spinning up images in another region and getting back online. Keep up to date AMI's in multiple regions etc. If you can be down a few days this is the easiest and is totally reasonable. Just remember with the growth of data this can start getting unrealistic.
As the acceptable time shrinks to operational from failure you need to have a more and more sophisticated plan. From simple hot standbys in other regions to complete multi-region duplication.
It really doesn't have to be complex at first, start with a simple backup/recovery strategy and then add as your requirements demand it.