No. A valid Tanzania visa covers both Mainland Tanzania and Zanzibar. An American applicant selects the first destination in the form and can then continue to the other part of the United Republic without purchasing a second Tanzania visa.
A common US itinerary begins at Kilimanjaro International Airport, continues through Tarangire, Ngorongoro or Serengeti, and ends with several nights in Zanzibar. Another begins in Zanzibar and later flies to Arusha. These are domestic movements within the United Republic of Tanzania for visa purposes. The application should identify the true first arrival point and present an itinerary that makes the sequence easy to understand.
Problems usually come from inconsistent evidence rather than the two-destination plan itself. For example, the form may say Zanzibar is first while the ticket lands at Kilimanjaro, or the hotel dates may overlap with a safari camp hundreds of kilometres away. A concise itinerary can solve this: list the arrival airport, first accommodation or tour operator, broad safari dates, Zanzibar dates and final departure. There is no need to upload a minute-by-minute schedule, but the main movements should be credible.
If the traveller leaves Tanzania for Kenya, Rwanda or another country and later returns, the re-entry should be considered against the Multiple Visa conditions and validity. The East African Tourist Visa used by some neighbouring states does not replace a Tanzania visa. Each country's permission remains separate, even when one tour company sells the entire regional trip as a single package.