Upon running the ProcessExplorer executable, it will extract the 64-bit version of the program to the AppData/Local/Temp folder and attempt to run from there. However, if you're like me and have restrictions on applications running from the Temp folder, this will cause the error. To get around it I simply navigated to my temp directory and move the procexp64.exe file to my Desktop and executed it from there. It opened right up and I was able to get back to what I needed ProcessExplorer for.
By default the AppData directory is hidden. The quickest way to get there is by clicking Start, then type %tmp% into the Search box and press Enter. Or type the path directly into the navigation bar, or choose to show hidden files.
Full path to the temp folder is C:\Users\"your username"\AppData\Local\Temp