After a Juvederm injection, how long until you can lay face down for a massage?

You should ask your injector, as they may have different guidelines. Generally, you should wait at least 24 hours or longer, depending on your treatment.  

I always advise my patients to wait at least 72 hours before adding any compression to the area. The guidelines can vary with area injected. Speak with your injector to answer the question specific to you,

You do not mention where the Juvederm was injected.  In general, I ask patients to wait 2 weeks after filler injections before having any other procedures such as facials, chemical peels or laser treatments to ensure any bruising or swelling has subsided.  This would also apply to lying face down for a massage.