Can you recommend an online telnet tool?

This one looks okay -
http://www.hq42.net/net_tools/test_telnet_server.php
 

I came across that one also, however they appear to be having problems because I can't open their website.

I can and it works.
This is their network tools homepage -
http://www.hq42.net/net_tools/index.php
 

Must be our firewall - I can access the website from a computer hosted on a separate network!

Seems to be really good, your firewall - at least for outgoing traffic     :-)

Problem solved - it is the virus scanner/web filter which is blocking my access to this website!

This website isn't quite what I am looking for. I am hoping to find a TELNET emulator which will allow me to send my own commands to our servers through a web server. For example, to test our Exchange Server I would first send the following TELNET command:

               telnet mail.bigpink.co.uk 25

Sorry, I'm not aware of such a site.

Sounds like you need to host a box externally and use nmap.

I use a combination of nmap (for manual tests) and telnet and nagios for automated tests from a box at home.  

I'm not aware of any websites that offer such a service, but you can run a linux server with a relatively low spec running nagios and nmap for free....

nagios - http://www.nagios.org
nmap - http://www.insecure.org

Nagios can test (at whatever frequency you want) service statuses, port statuses, pingable hosts - the options are vast.

nmap can test whether ports are open/closed/filtered and also fingerprint the software versions of listening services on the ports (nmap -sS -sV mail.yourdomain.com -p 25)

Old I know, but still the 5th result on google. I found http://telnet-online.net/ and seems to be what you were looking for. This is mainly for others who stumble onto this thread.