I'm not familiar with the OAuth API, but Dispatch allows you to add arbitrary HTTP Request headers with the <:<
method/operator.
So mashing together your code example above and this "Authorizing a request" example from Twitter's Developer site, you might get something like this:
val authentication = """OAuth oauth_consumer_key="xvz1evFS4wEEPTGEFPHBog", oauth_nonce="kYjzVBB8Y0ZFabxSWbWovY3uYSQ2pTgmZeNu2VS4cg", oauth_signature="tnnArxj06cWHq44gCs1OSKk%2FjLY%3D", oauth_signature_method="HMAC-SHA1", oauth_timestamp="1318622958", oauth_token="370773112-GmHxMAgYyLbNEtIKZeRNFsMKPR9EyMZeS9weJAEb", oauth_version="1.0""""
val request = url("https://api.twitter.com/1.1/search/tweets.json?q=%23%sresult_type=mixed&count=4" format w.queryValue)
val authHeader = Map("Authentication" -> authentication)
val requestWithAuthentication = request <:< authHeader
val response = Http(requestWithAuthentication OK as.String)
I haven't verified whether this actually works, but hopefully it should get you going in the right direction.