Using a forum as a support channel is a nice addon to your company, but it requires a lot of hard work to keep going.
Like many others have said, a dead forum is bad publicity for your company. On the other hand, a working, active forum is one of the best ways to make a company "famous" along its customers and even potential customers.
So it goes both ways. If you decide to keep it, make it look active. Post offers or giveaways so that your customers comes back, even if they don't have a problem.
Also make sure that the forum is NOT a replacement for your helpdesk/ticket system. Only a place to get general help from people who also like what they are doing, being a customer of you