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This happens because of OVER_QUERY_LIMIT. The plugins uses Google API to get the driving distance. When running this feature to many times in a short time it returns OVER_QUERY_LIMIT error instead of the distance.
One way to maybe overcome this issue is to place a delay of 2 seconds before each query call ( which will also display the distance in a delay after page load ). However, this feature is not recommended for use ( as mentioned in the form editor ) and I am most likely going to deprecate it in the next update of the plugin. This has been an issue for many users and if your site reaches OVER_QUERY_LIMIT very often it be can banned by Google.
My plane is to remove this feature from the search results and add it to the Single Location widget so visitor will be able to see the driving distance when viewing a specific post.
My advice to you is to remove the driving distance from the search results for the moment.