Do desktop computers catch fire often?
In 2007-2011, there were an estimated 730 reported U.S. home structure fires involving office equipment per year...
[of these] Computers [accounted for] 48%
In 2007-2011, there were an estimated 630 reported U.S. non-confined non-home structure fires involving office equipment per year...
[of these] Computers [accounted for] 36%
So a little calculation: 48% of 730 + 36% of 630 = 350 + 227 = 577 computer fires per year. [Note that this category is a bit confusingly worded, I'm not sure what constitutes "office equipment" but I couldn't find computers in any other category, so they may all be accounted for in this one.]
So computers seem to catch fire on average about once every 15 hours in the US. (2007-2011)
Is the risk non-negligible?
During the five-year period of 2007-2011, U.S. fire departments responded to an estimated average of 366,600 home structure fires per year. ...
Leading Causes of Home Structure Fires: 2007-2011
Cooking equipment 43%
Heating equipment 16%
Electrical Distribution and Lighting equipment 6%
A little more math:
43% of 366,600 = 157,638 US home fires caused by cooking equipment
48% of 730 = 350 US home fires caused by computers (see above)
350/157,638 = 1 : 450
You are 450 times as likely to have a (reported) US home fire from cooking equipment than a computer.
Maybe you could just eat out one more day per year instead of cooking, this should satisfy your safety conscious room-mates by mitigating the added risk of an extra computer.