I am considering using the Gunning Fog Index and the Flesch Reading Ease to calculate variable information for my dissertation. Do I need to obtain permission from someone to use these formulas? In other words are they considered to be copyrighted data collection instruments?
It's very doubtful that you'd need permission to use either formula. In any case, I'm not sure who you'd even ask. The formulae are simple, unpatented, and widely published, and both Gunning and Flesch are dead.
Both formulae were published by their developers, who actively encouraged their use (Gunning in his 1952 book The Technique of Clear Writing, Flesch as quoted by the University of Canterbury, NZ). The formulae are part of the existing literature, and it was clearly the authors' intention that these formulae be disseminated and used for relevant purposes.
The formulae, among others, are implemented in several pieces of software (including copyright-averse free software like LibreOffice) and online calculators. Even if there were some restriction on their use (which there doesn't appear to be), it is clearly unenforced, and hasn't been for a very long time.
It may, however, be appropriate to cite the work of Gunning and/or Flesch, depending on the norms of your field.