To remove the white rings left by cups or glasses, rub the wood with a mild abrasive. First, try cigarette or cigar ashes moistened with a little cooking oil; apply with your finger. If that doesn't work, try successively stronger steps: use table salt and a drop of water; silver polish or car polish; and, finally, on teak, use fine glass paper and then oil.
Erase superficial ink spots with oiled pumice or steel wool.