Essentially, your customer cannot use the product/data without a license. As mentioned above, an evaluation might be appropriate if the customer only needs this for a short time. Also, you can export your issues to csv, which will provide the data in a format that can be imported into most issue tracking software, which might be a more portable solution.
Finally, you might consider setting your permissions so that you can create a single user for that customer, and restrict them to that particular project (also see Managing Project Visibility), so that they can view this information as long as you have the OnDemand product.
Hope that helps a little!
You can backup the entire JIRA as explained here: https://confluence.atlassian.com/display/AOD/Exporting+issues#Exportingissues-BackingupandexportingJIRAOnDemanddata
and then import it into a local JIRA instance of the same or higher version.
How long will they need the information? If it is short term, I would suggest using a demo license, perform a backup/restore into the demo environment and then hack everything but the project out. Exporting just a project with complete definitions, context, attachments, comments, states, etc. is problematic with out a lot of back end scripting. You can then publish the link to just that user and open up the project for viewing.
If they will need a longer term beyond what the temp license provides, or they "want to use the instance", well...they will need to license that and you'll want to sanitize your configurations.
If you do that, make sure you disable mail polling until you've cleared everything out and disabled any reports folks have setup to be mailing themselves of the off chance that where you setup/restore will have open smtp access.
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