Can the OD5000 measure the thickness of thin, transparent tapes?

This device cannot solve this task due to its technical performance limits. 
The OD5000 won't be able to separate the two signals (front site and back site) of a thin tape properly. A confocal sensor such as the OD7000 is the right device for this application. 

Please note that the specified minimal thickness the OD5000 can measure (see operating instructions) are related to crown glass. Transparent plastic foils or tapes do not have the same material properties. Because of this it is not possible to reach these specified values on such materials.