Limesurvey doesn't have this option at the moment. It sounds like something simple, but I don't think it is that simple, because you would have to add an additional dynamic filter etc.
For now what you could do, if you don't just export everything and then filter it in Excel or something, is check the IDs that are within this date range and than filter it via the IDs. But I guess it would be a lot simpler to just export everything and then filter in an external analysis software (as simplest solution Excel).
Now, for the future you could create a feature request. I don't know what the chances are that this will be implemented (depends on dificulty, "felt" necessity, etc. I guess):
www.limesurvey.org/en/community-services/feature-requests
I answer at the LimeSurvey forum in my spare time, I'm not a LimeSurvey GmbH employee.
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Thanks, i've sent in a quote request for custom development of this feature. Looking forward to the response on that one so we can get this developed quick, if not too expensive.