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By Date Picker I believe that you mean the DateTime field found in the Quick Tools menu.
For the above we must do several things and I will take you through all the steps.
1. Disable Past dates.
To do this you must click on the little gear wheel of the DateTime field and then select Properties
Once there, you will see the option to turn the past dates off.
2. Now we need to limit the dates
For this we stay in the Properties and scroll all the way down where we can see the button for Limit Dates wizard.
*Now, we can disable the past dates in here as well, I like to do it in both places :)
So to do it here, we just uncheck the Past here and add today>today+3 in the custom date range field to limit the range that we do not want to see as available.
Once you save and run your form, you should be able to see it working.
Do let us know if you have any questions along the way and we would be happy to assist.
I just wanted to let you know that this is the link to the form that has this set: http://form.jotformpro.com/form/51807037811957, but at this time it seems that there is happening within the limit dates option.
I have however notified our developers in regards to this so they will take a look at the issue and resolve it so it will be available shortly again.