Can Mathematica delete some of the 0's from a list?

The easiest way would be to use

{1, 2, 3, 0, 0, 4, 5, 0, 0, 0} /. {a___, 0 ...} :> {a}

/. is synonymous with ReplaceAll. It tries to replace values on the left hand side with the rules on the right hand side. In this case {a___, 0...} is the pattern; a___ matches zero or more elements, and 0... matches zero or more zeroes. :> {a} takes the a that corresponds to the matched expression and returns it.

The TakeWhile and replacement based answers will be very slow with large lists and/or large lists with many trailing zeroes.

Something like

dropper=With[{s = Split[#]}, If[s[[-1, 1]] == 0, Join @@ (Most@s), #]] &;

will be ~2000X faster on a list of 50K length vs a replacement solution, advantage growing with size. There are other even faster methods for really huge flat numeric lists... e.g.:

dropper2 = 
 With[{s = SparseArray[#, Automatic, 0]["NonzeroPositions"]}, 
       If[s === {}, {}, #[[;; s[[-1, 1]]]]]] &;

Using:

test = Join[RandomChoice[{1, 10} -> {1, 0}, 100000], ConstantArray[0, 100000]];

The SparseArray is ~10X faster than dropper, which is itself ~15X faster than the TakeWhile, which is itself ~365X faster than the replace-based solution, making the SparseArray ~45,000X faster for this test...

Technically, @Bill's and @Pickett's clean answers do not quite delete the 0s after "the last positive integer." A teeny alteration fixes Bill's answer:

{1, 2, 3, 0, 0, 4, 4.2, 0, 0, 0} /. {a___, b_Integer /; b > 0, 0 ..} -> {a, b}