The direct answer for almost all teams is that QA must be complete to call a user story done. I would take that further and say that I have never worked with a team where this was not true.
The More Thorough Answer
In Scrum, a product increment (the product as it stands at the end of the sprint) must be "potentially shippable". This is a confusing term, but what it means is this:
The decision to ship a product to end users or not should be a business decision (do the available features provide a significant enough value to the end users that it is worth the cost to give it to them and the stress of them adopting the new version of the software). If there are technical reasons why the product can't be shipped (bugs, not QA'd, not in a deployable state, etc), then it is no longer a business decision. Therefor, a product increment is considered potentially shippable if there are no technical reasons that the product cannot be shipped to end users.
The "Definition of Done" acts as a list of those items that must be completed in order to meet the criteria for a Potentially Shippable product. So, the more complete answer to your question would be: If QA tests must run and pass before your product can be shipped to end users, then it should be in your definition of done and must be completed in the sprint in order to move those backlog items to a "Done" state.