Industrial strength velcro peeling off MDF
Super glue might work, although my first thought was to finish the mdf with polyurethane first. This will seal the mdf and allow the adhesive to remain sticky for a long time.  Rubbing superglue on the area before the velcro goes on might get the same result too if you don't want to finish the whole thing. 

I think this is what's going on : the mdf is very porous and absorbent. For example, it soaks up finish rapidly until it's saturated and only then can you build up finish thickness. I think the mdf is basically extracting the lower molecular weight components out of the velcro adhesive, which are the that make it tacky. With those gone, the adhesive is now more brittle and fails in short order.