You didn't say what the record key for each record is, so I'm just adding the records to an overall array. I'm also not sure what the numbers inside the tags are for, so the following assumes that each tag is a key with no modifications.

use strict;
use warnings;
use Data::Dumper;

my ($temp, @results) = '';

### Fast forward past header
while (<DATA>) {
    last if m/<eoh>\s+$/;
}

### While there are records remaining...
while (<DATA>) {
    $temp .= $_;

    ### Process if end of record tag reached
    if (m/<eor>\s+$/) {
        my %hash;
        $temp =~ s/\n//g;
        $temp =~ s/<eoh>.*//;
        $hash{$1} = $2 while $temp =~ /<(.*?)>([^<]*)\s/sg;
        push @results, \%hash;
        $temp = '';
    }
}

print Dumper(\@results);

__DATA__
Exported by jLog (c)2006 LA3HM, V 3.90.2.7 according to ADIF <adif_ver
+:1>2 <PROGRAMID:4>jLog For jLog info: mailto:mail@jlog.org http://jlo
+g.org/ Proposed ADIF2 Extensions may be included <eoh> 

<qso_date:8:d>20051029 <time_on:6>213400 <call:4>VC3O <band:3>20M <mod
+e:3>SSB <operator:3>VHF <rst_sent:2>59 <rst_rcvd:2>59 

<dxcc:1>1 <stx:1>4 <srx:1>4 <ituz:1>4 <cqz:1>4 <pfx:3>VC3 <cont:2>NA <
+freq:2>14 <qsoComplete:1> <app_jlog_qso_number:4>0001 <app_jlog_eqsl_
+qsl_sent:1>Y <app_jlog_eqsl_qsl_rcvd:1>Y <app_jlog_lotw_qsl_sent:1>Y 
+<qsl_sent_via:1>E <eor> 

<qso_date:8:d>20060701 <time_on:6>183206 <call:5>VE6GG <band:3>20M <mo
+de:3>SSB <operator:3>MWB <rst_sent:2>59 <rst_rcvd:2>59 

<dxcc:1>1 <stx:2>27 <srx:2>AB <ituz:1>2 <cqz:1>4 <contest_id:3>RAC <pf
+x:3>VE6 <cont:2>NA <freq:8>14.16299 <state:2>AB <qsoComplete:1> <app_
+jlog_qso_number:4>1257 <app_jlog_eqsl_qsl_sent:1>Y <app_jlog_eqsl_qsl
+sdate:10>2006-07-01 <app_jlog_lotw_qsl_sent:1>Y <app_jlog_lotw_qslsda
+te:10>2006-07-01 <operator:4>N7DQ <eor>
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