When should one use final for method parameters and local variables?

Obsess over:

• Final fields - Marking fields as final forces them to be set by end of construction, making that field reference immutable. This allows safe publication of fields and can avoid the need for synchronization on later reads. (Note that for an object reference, only the field reference is immutable - things that object reference refers to can still change and that affects the immutability.)
• Final static fields - Although I use enums now for many of the cases where I used to use static final fields.

Consider but use judiciously:

• Final classes - Framework/API design is the only case where I consider it.
• Final methods - Basically same as final classes. If you're using template method patterns like crazy and marking stuff final, you're probably relying too much on inheritance and not enough on delegation.

Ignore unless feeling anal:

• Method parameters and local variables - I RARELY do this largely because I'm lazy and I find it clutters the code. I will fully admit that marking parameters and local variables that I'm not going to modify is "righter". I wish it was the default. But it isn't and I find the code more difficult to understand with finals all over. If I'm in someone else's code, I'm not going to pull them out but if I'm writing new code I won't put them in. One exception is the case where you have to mark something final so you can access it from within an anonymous inner class.

• Edit: note that one use case where final local variables are actually very useful as mentioned by @adam-gent is when value gets assigned to the var in the if/else branches.

Is it something I should make an effort to remember to do?

No, if you are using Eclipse, because you can configure a Save Action to automatically add these final modifiers for you. Then you get the benefits for less effort.

The development-time benefits of "final" are at least as significant as the run-time benefits. It tells future editors of the code something about your intentions.

Marking a class "final" indicates that you've not made an effort during design or implementation of the class to handle extension gracefully. If the readers can make changes to the class, and want to remove the "final" modifier, they can do so at their own risk. It's up to them to make sure the class will handle extension well.

Marking a variable "final" (and assigning it in the constructor) is useful with dependency injection. It indicates the "collaborator" nature of the variable.

Marking a method "final" is useful in abstract classes. It clearly delineates where the extension points are.