Lots of people have suggested good ways to do it, but here are a few points about your original version.

perl -i -e 'while(<>){print $_[3];}' file.txt
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The -i switch says that you want to edit in place. Normally you use that like this: -i.bak. The .bak (or whatever else) becomes a suffix for a copy of your original file. The original becomes file.txt.bak and file.txt gets edited. If you use -i by itself without a suffix, then you don't get a backup. That's dangerous. Since you're not editing at all here, you don't want the switch. (In this example, if you had successfully done print if $. == 4 and used the -i switch with no suffix, you would get no backup and only line 4 left in your original file. I don't think that's what you wanted.)

The -p and -n switches are worth looking at in perlrun since they will give you the while(<>) loop for free.