No, it will not affect your credit score.
This is because Whitepages TenantCheck is set up specifically so that you as the prospective tenant are requesting the release of your own background and credit info. This counts as a soft inquiry and does not impact your credit score.
If a landlord were to initiate the credit check on their own, this would result in a hard inquiry on you, the applicant's, credit report and can have negative effects on your overall credit score. This is similar to the process you would see when applying for a home loan or credit card.