NO, you should not install antivirus software on a web server, at least not as your first choice. You should install a Host Intrusion Detection System (HIDS) like Samhain, Tripwire or the like.
The question is a duplicate of this older question, please see my answer there. The gist of it is that HIDS has a better detection rate against hacks than antivirus (AV) software.
If you enable uploads of Office types of documents, i.e. PDFs, Word docs and other documents that can embed active content, then a normal antivirus package would make sense IMHO, to scan for macro viruses in the uploaded content.
This answer assumes that your web server is completely separated from your internal office network -- which should be a obvious security practice under all circumstances.
Edit: To clarify, AV follows a 'blacklist' approach, i.e. it has signatures for a known list of 'bad' programs, and alerts you (and optionally takes action itself) when it sees one of these 'bad' programs on your server.
HIDS follows a 'whitelist' approach, i.e. it has a list of all the programs on your server that should be there, and alerts you whenever executable code is added or changed without your approval. This approach will have a better detection rate against one-of-a-kind hacks and zero-day exploits, at the expense of (many) more alerts being given, especially if you don't take the time to configure the HIDS optimally.