Are there any hosting companies that allow you to rent server space and install Windows programs that can be left to run 24/7? I could do this on my home machine but don't want to leave it on permanently due to fire risk.

Why would it be a fire risk? LIR

I heard it was. Some computers, if you leave them on permanently, can be a fire risk. Fans break down, sometimes overheat, CPU, electrical fire.

Well, consider that the same could happen with any electrical device. One of the worst offenders is a shorted toaster. Do you unplug it after every use? I have always left my PC's run 24/7 and have been doing so for over 15 years. Total number of electrical fires caused by my PC's = 0 (zero)

There are plenty of companies that offer vpn access to either dedicated servers or virtual servers. I never leave computers run at home either. Data centers are built to do just what you want. "Best Practices", Event viewer, host file, perfmon, antivirus, anti-spyware, Live CD's, backups, are in my top 10

They all seem to be web server hosting companies though. Noone actually lets you install a program on Windows do they ?

Take a look at ServerBeach. They have what you're looking for - dedicated servers. These are dedicated boxes running Windows 2003 or Windows 2008 that are in one of their data centres. They also look after the hardware for you. I've been using them for a while now and am very impressed. You can see the various hardware configurations at http://www.serverbeach.com/catalog/cust_ref_landing_new.php?REF=PACUYK2FA9