Couple of comments here:

1. I admit that the discrepancy between what happens with $var3 and $var4 is bizarre. I can't figure out why one is allowed and one isn't.
2. On the other hand, a here doc is kind of like a string literal. You're effectively trying to modify a string literal without assigning it to a variable first. This is dangerous even if it does sometimes work.
3. Why are you applying map to a single string? map is meant to work on lists, and a here doc is only a single string, that is a scalar, that is a list of length one. Just assign it to a variable and then use s///. You've already shown how to do that in a single line of code (i.e. (my $foo = <<EOT) =~ s/foo/bar/).
4. Update: You say in your comments that map returns the number of substitutions. That's not true. That's what s/// returns in scalar context. map in scalar context returns a count of the elements in the list, which in your case is always 1.