it is a hard one to know the answer to. Just about every automotive repair site, and a lot of other metal and woodwork forums have sections for homemade tools, but as you say, it costs money to keep it all up and running.
Charging is the quickest way to lose people who are "browsing" - similar to when I walk in a store, if the sales persons keeps hounding me to buy something, it just makes me defensive and unwelcome - it's as though all they want is my money. HMT is about people sharing ideas on tools they make/ modify. It has a feel of people all contributing, so anything which made that feel change to one of "give me some moeny to look at this" starts to rankle on some people.
I'll also add, never won tool of the week, not sure if I ever would. If the bar of entry is too high, it will be harder to get fresh blood, and the forum could easily become an echo chamber for the group who already have been here a while, know what it takes to win tool of the week, etc.
If I was looking to reduce costs, I wouldn't give physical awards for tool of the week - bragging rights are enough, or one off discounts at the store.
my 2 cents,