I am using Apache Solr Search on my site, and would like links on the search results page to open in a new window/tab. How can I accomplish this?
You should install Custom Search module as it does provide an option to open results in new tab or lightbox.
Otherwise you can always look into using custom functions or js to achieve the same in your search template.
Since you are using solr and search-result.tpl.php you should have a line like:
<a href="<?php print $url; ?>"><?php print $title; ?></a>
change it by adding target value like:
<a href="<?php print $url; ?> target='_blank'><?php print $title; ?></a>