how old is the car? The reason i asked this is because in the the adhesive of the wood trims(double sided tape) may have already hardend and has already "eaten through" the vinyl, leaving a permanent mark on the dash and sidings.
in that case just leave it as is. baka kasi sumama na yung vinyl ng dash pag tinanggal yung wood trim. and if tangalin, iba na color ng panel na pinagtangalan ng trim from the rest of the dash. its just like removing emblems sa paint, after you remove the emblems on an old car may mark na, yung shape nung emblem... in some cases, if you look closely may pin hole sa paint na pinagtanggalan ng emblem... or in worst cases sumasama na yung paint sa emblem.
dati lancer pizza ko yung aircon panel, wood trim din. Aalisin ko sana kaso risk ko is baka iba na color from the rest of the dash. lalo lang lalaki problem ko.