if ( $data eq "8000" || $data eq "0080" ) {
    $tid = "TEST";
    $buf = "TEST SYSTEM";
} 
elsif ...
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and

if ( $data eq "8000" or $data eq "0080" ) {
    $tid = "TEST";
    $buf = "TEST SYSTEM";
} 
elsif ...
[download]

are strictly equivalent (in this case). As others have commented, || and or have different precedence, but that doens not make any difference here. Personally I prefer to use the lower precedence or when grooping to Boolean expresions, but that's a matter of taste and that's debatable.

if ( $data eq "8000" || "0080" )
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on the other hand is just wrong, although the compiler will assign a meaning to it. It is wrong in the sense that it is simply not doing what you intend: this is saying : if $data is "8000" or if "0080", which will be always true, because "0080" is a true value. So fix it as you said (with either of the two syntaxes shown at the beginning of this post) and your program should work properly.