Are you clicking on the little calendar icon in the inspector field, then clicking on a date, then closing the inspector by clicking on the red dot? That would cause your change to be lost. Hit return to save your change.
The idea behind the Next Review date is that you bring up a Project mode view, grouped by Next Review. The top of the window will be the project most due for review (oldest Next Review date). You look at the selected project, and then use the Mark Reviewed button (not in the default toolbar, but available for addition if you use View->Customize Toolbar, or press cmd-shift-R) to mark it reviewed. That advances the Next Review date by the Review Interval, updates the display, and selects the next project. You continue through all of the projects due for review today or earlier, then stop. Doing this every day will allow you to have projects with differing review schedules depending on how often they need attention; this is a very powerful feature, and with a bit of cleverness, you can actually spend only a small amount of time each day on reviewing, instead of doing everything in one indigestible lump as you would typically do in a canonical paper-based system.
If my guess about what you're doing above isn't correct, I would select the project that you can't set the next review date on and do the Mark Reviewed operation, then see what happened to the next review date. If it didn't change, try File->Rebuild Database and repeat the test. If still no luck, contact the Omni support people tomorrow via Help->Send Feedback or by telephone (800-315-OMNI or +1-206-523-4152).