As Tru said, in CloneDVD, on the subtitle page, make all your subtitle choices in the right hand pane (entitled "Stream Preferences"), not the left hand pane (entitled "Stream Configuration"). Do not even click in the left pane area. But visually check that none of the sub-tracks are "de-selected" in the left hand pane. This was a small issue before version 126.96.36.199 was introduced but is supposed to be resolved now. But try it anyway.
In CloneDVD, did you choose "Preserve Menus"? Some movies are mastered in such a way that they matrix the audio and subtitle tracks through the menu structure and a backup will not play correctly if you did not include the menus.
Check your DVD player's onscreen setup settings. Are subtitles set to completely "OFF". Different brands of players have various different subtitle settings choices. Try different settings. This, however, is the last possibility I listed for you because it is probably not the issue since I infer from your wording that the original does play these forced suptitles.