A check of the source code reveals that the SHIFT method for Tie::File is implemented in terms of the SPLICE method:

sub SHIFT {
  my $self = shift;
  scalar $self->SPLICE(0, 1);
}
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Besides, the file is updated after every shift operation which means you'd be re-writing the file four times (!)

I don't think Tie::File is the right approach to manipulate 700 MB log files.

Update: I have a feeling the OP is running into out of memory problems because Tie::File is keeping track of the start of each line even though it doesn't need to. For a multi-megabyte file this clearly would be a problem. However, this is currently just a conjecture.