As Rolf said, you want to print the entire array, not just the blank line at the beginning. Better yet, remove blank lines and also perhaps the extra blank after the keyword?

use strict;
use warnings;

my $string="TOPO_DOM 1; /note=Cytoplasmic; /evidence=ECO:0000269|PubMe
+d:24744376; TOPO_DOM 25..74; /note=Extracellular; /evidence=ECO:00002
+69|PubMed:24744376; TOPO_DOM 100..115; /note=Cytoplasmic; /evidence=E
+CO:0000269|PubMed:24744376; TOPO_DOM 141..159; /note=Extracellular; /
+evidence=ECO:0000269|PubMed:24744376; TOPO_DOM 183..227; /note=Cytopl
+asmic; /evidence=ECO:0000269|PubMed:24744376";

my @split_topodom = split(/TOPO_DOM\s*/, $string);
@split_topodom = grep {/\S/}@split_topodom;  # remove blank lines
print join "\n",@split_topodom;

__END__
1; /note=Cytoplasmic; /evidence=ECO:0000269|PubMed:24744376; 
25..74; /note=Extracellular; /evidence=ECO:0000269|PubMed:24744376; 
100..115; /note=Cytoplasmic; /evidence=ECO:0000269|PubMed:24744376; 
141..159; /note=Extracellular; /evidence=ECO:0000269|PubMed:24744376; 
183..227; /note=Cytoplasmic; /evidence=ECO:0000269|PubMed:24744376
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Update: Of course you can combine the grep that removes blank lines into the split line.
my @split_topodom = grep{/\S/}split(/TOPO_DOM\s*/, $string);