If you are going to paint it you normally do not need to remove it...just a light sanding to help the primer bind. The only reason to remove it would be if it is so damaged that it is falling apart of coming off in chunks.
Depending on how the veneer was originally applied there may be adhesive ridges which should be smoothed for flat paint finish.
Debra, one thing we have had good results with is an iron and a paint scraper. Heat the veneer with an old iron to release some of the adhesive or glue and gently scrape the veneer off. Depending on the piece, it can be time consuming but is worth the effort. Good luck!
Thank you for your info.