Are you converting the movies to dvd or Blu-ray, or are you just putting the file on a DVD?
Are you playing it on a PC or a standard DVD/Blu-Ray player?
When you play the movie on the PC from the download (not from the burned disc) are the adds there?
The most likely thing is that the movie you downloaded has ads built into it; you cant remove the ads after-the-fact the only solution is to get legal legit movies.
Possible your burner/encoder program is slipping in ads; solution is to pay for decent software or find a different freeware one.
If playback is on a PC and not a standard home theater dvd-player then its possible the PC is just infected (or the program you are viewing the video on just has ads).