Subfloors for tile MUST be rigid, usually a min. of 1 1/4" is needed or you will have cracking tiles and/or grout. You don't need to remove baseboard, but do remove the 1/4 round, then install new after done. You could remove the baseboard, then install new basebaord over the tile edges for a cleaner look, but 1/4 round is the usual way.