I have a domain which previously belonged to someone hosting a forum on it.
Can I use that traffic to give my domain a boost, by taking the 404 traffic and 301 redirecting it to an existing page of mine?
Yes. For a period. However, I do warn you that if link keywords do not match the target page, these links can be downgraded and Google, depending upon the number, can consider this as spam. If your site is the same basic topic as the forum, then perhaps you can match the link keyword topics to relevant pages to prevent this.