When Marsha O’Loughlin goes to Paris this March, she won’t be snapping photos of the Eiffel Tower. She’ll be too busy running up its stairs.

The 75-year-old is one of 131 participants who plan to compete in the 2020 Verticale de la Tour Eiffel, a race up 665 of the tower’s stairs.

Ms. O’Loughlin participates in a sport called tower running,...