There are no copyright notices on the first-edition dust jacket nor the second-edition dust jacket (both of which have the same front cover art) as shown in the links below.

First edition, 1982

nocloo.com:

• image 1 (front cover)
• image 2 (front and back covers, spine)
• image 3 (front and back flaps)
• image 4 (book without dust jacket)
• image 5 (book without dust jacket)
• image 6 (book without dust jacket)
• image 7 (book without dust jacket)
• image 8 (inside front cover)
• image 9 (title page)
• image 10 (copyright page)
• image 11 (pages)
• image 12 (pages)

Second edition, 1984

Biblio:

• image 1 (front cover)
• image 2 (book without dust jacket)
• image 3 (book without dust jacket)
• image 4 (title page)
• image 5 (copyright page)
• image 6 (pages)
• image 7 (dust jacket)
• image 8 (insert/catalog)

Using the search term "Dark Tower" and looking at the year 1982, we see that there are two possibly relevant copyright registrations: TX0001057290 and TX0001057289.

TX0001057290 says "Copyright Claimant: Michael Whelan on ill. in book;" and "Edition: 1st ed."

The registration does not mention the dust jacket. Furthermore, even if the registration covers the dust jacket, the same front cover art was subsequently published in the second-edition dust jacket without notice, which would risk forfeiting the copyright for the cover art.

Using the search term "Dark Tower" and looking at the year 1984, we can see that there are no copyright registrations relating to the second edition.

TX0001057289 refers to the literary work and not the dust jacket.

Thus the front cover art is in the public domain in the US.

This work is in the public domain because it was published in the United States between 1978 and February 1989, inclusive, without a copyright notice, and its copyright was not subsequently registered with the U.S. Copyright Office within 5 years. For further explanation, see Commons:Hirtle chart as well as a detailed definition of "publication" for public art after 1977. Unless its author has been dead for several years, it is copyrighted in the countries or areas that do not apply the rule of the shorter term for US works, such as Canada (70 pma), Mainland China (50 pma, not Hong Kong or Macau), Germany (70 pma), Mexico (100 pma), Switzerland (70 pma), and other countries with individual treaties. See this page for further explanation. Deutsch ∙ English ∙ español ∙ français ∙ italiano ∙ 日本語 ∙ 한국어 ∙ македонски ∙ português ∙ português do Brasil ∙ русский ∙ sicilianu ∙ slovenščina ∙ 中文（简体） ∙ 中文（繁體） ∙ +/−