Will using an adhesive remover on the headliner board & wiping it clean cause the new adhesive not to stick?

No it should not make a difference in the new adhesive making a good bond, if you let it dry before you apply the new adhesive. That has been our experience with adhesive remover such as Acry-Solv anyways - but the Acry-Solv adhesive remover is a professional grade product and it always makes a difference using high quality products.