What is the ruling on visiting the Dead Sea for tourism or treatment, and visiting the city of Petra?

There is no impediment to visiting the Dead Sea and Petra as long as there is no Shar'i (Islamic legal) evidence proving that they are among the abodes of those who were punished (mu'adhdhabīn). The basic principle regarding things is permissibility. The prohibition against entering the abodes of those who were punished, unless the Muslim is weeping, humbled, and taking heed from what befell them, applies when the motive for going is not admonition and contemplation. However, if the purpose of going is admonition and contemplation, then there is no harm in it. Indeed, Allah has commanded traversing the earth to take heed and observe the fate of those who denied the truth.