The Collaborative International Dictionary of English v.0.48:

lap up \lap up\, v. t. [See lap, v. i.]
   1. To take up (drink or food) with the tongue; to drink by
      licking up.
      [1913 Webster]

   2. (fig.) To accept or enjoy enthusiatically and
      uncritically; as, to lap up praise.
      [PJC]

WordNet (r) 3.0 (2006):

lap up
    v 1: take up with the tongue; "The cat lapped up the milk"; "the
         cub licked the milk from its mother's breast" [syn: lap,
         lap up, lick]

Moby Thesaurus II by Grady Ward, 1.0:

48 Moby Thesaurus words for "lap up":
   and sinker, be a sucker, be taken in, bib, bite, devour,
   drain the cup, dram, drink, drink bottoms-up, drink deep,
   drink hard, drink off, drink up, eat up, fall for,
   follow strong drink, gargle, go for, gobble up, grog, gulp down,
   guzzle, imbibe, knock back, lap, lick, line, nibble, nip, quaff,
   sip, slurp, soak, soak up, sponge up, sup, swallow,
   swallow anything, swallow hook, swallow whole, swing at,
   take the bait, tipple, tope, toss down, toss off, tumble for