Short answer - don't.
Long answer - Rewrite a whole swathe of the core functionality. The resolution field is embedded deep into the structure of the system. It's used as the baseline "is this issue open or closed" question, and excising it means removing that and assuming all issues are always open, or replacing it with something a lot more complex that can handle the open/closed thing.
I agree with Nic, you shouldn't remove the 'Resolutions' from JIRA. Several gadgets won't count the issues as, in fact, resolved, and you may face problems when estimating time, creating graphics, etc.
If you want it anyway, just remove the 'Resolutions' field from the 'Resolve Issue Screen'.
The resolution field is a system field that controls some central functionality in JIRA.
The most important one (in my opinion) is that the field tells JIRA when an issue is "done". This is usually shown as a strikethrough issue-key. Same gadgets are also dependent on this.
If you don't need this field, which I think is odd since you can use this "definition-of-done" in JIRA in boards etc in a very neat manner, you should just make sure it is removed from all screens in your screen schemas. You could also hide it in your field configurations.
But if you do this you will not be able to use some of the functions in JIRA such as the Issues in Progress gadget and Assigned to Me gadget. Both of these utilize the resolution field.
Hope this helps!
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