VBA MS Word - How do I find a tab that's only at the beginning of a line and replace it with nothing?

Try this. The first paragraph is treated separately.

Sub RemoveLeadingTabs()
    If ActiveDocument.Range(0, 1).Text = vbTab Then
        ActiveDocument.Range(0, 1).Delete
    End If
    With ActiveDocument.Content.Find
        .ClearFormatting
        .Replacement.ClearFormatting
        .Text = "^p^w"
        .Replacement.Text = "^p"
        .Forward = True
        .Wrap = wdFindStop
        .Format = False
        .MatchCase = False
        .MatchWholeWord = False
        .MatchWildcards = False
        .MatchSoundsLike = False
        .MatchAllWordForms = False
        .Execute Replace:=wdReplaceAll
    End With
End Sub

A more comprehensive approach is to use a wildcard Find/Replace, where:

Find = ([^13^l])^t

Replace = \1

The other solutions you've been offered work only if the new line is preceded by a paragraph break; the above handles manual line breaks as well. As a Find/Replace, neither the above nor Suzanne's will do anything with the first paragraph in the document. For that, you need a macro like the one Hans posted:

Sub RemoveLeadingTabs()
Application.ScreenUpdating = False
With ActiveDocument.Range
  .InsertParagraphBefore
  With .Find
    .ClearFormatting
    .Replacement.ClearFormatting
    .Text = "([^13^l])^t"
    .Replacement.Text = "\1"
    .Forward = True
    .Format = False
    .MatchWildcards = True
    .Wrap = wdFindContinue
    .Execute Replace:=wdReplaceAll
    End With
    .Characters.First.Text = vbNullString
End With
Application.ScreenUpdating = True
End Sub