Previous solutions have suggested redirecting STDOUT, which would work, but consider using the built in one-argument select statement:

$oldFH = select(NOTHING);
do "filename.txt";
print "Won't Print\n";
select($oldFH);
print "Will Print\n";
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The select statement will change the default filehandle for all manner of things, including $| and print. It also means that if a called module does something like:

print STDOUT "This will print";
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... then it will print. However I'd be surprised if any module actually does this. It also gives you an 'out' if you want to force something to print while you have the select redirected.