Moving files to the Trash with Automator

The Move Finder Items to Trash action should do what you want, if you are providing the complete file path. How are your files defined in the text file (name only, POSIX path, etc)?

Since your text file just has the names, an action is needed to build the complete file path. Most of Apple's text and file actions don't use variables, but a Run AppleScript action can be used to fill in the blank. Give this workflow a try (I haven't tested it with a large number of files, so I don't know how slow it will be):

1) Ask for Finder Items {Files} -- choose the text file
2) Combine Text FIles
3) Filter Paragraphs {Return paragraphs that are not empty}
4) Run AppleScript -- paste the following script:
<pre style="
font-family: Monaco, 'Courier New', Courier, monospace;
font-size: 10px;
font-weight: normal;
margin: 0px;
padding: 5px;
border: 1px solid #000000;
width: 720px;
color: #000000;
background-color: #FFEE80;
overflow: auto;"
title="this text can be pasted into an Automator 'Run AppleScript' action">
on run {input, parameters} -- build file paths from a list of names

set output to {}
set SkippedItems to {}

set TheFolder to (choose folder with prompt "Select the folder containing the files:") as text

repeat with AnItem in the input
try
set AnItem to (TheFolder & AnItem)
get AnItem as alias -- test
set the end of the output to the result
on error
set the end of SkippedItems to AnItem
end try
end repeat

if SkippedItems is not {} then
display alert "Error with AppleScript action" message ((count SkippedItems) as text) & " items were skipped (file not found)"
end if

return output
end run</pre>
5) Move Finder Items to Trash