Use the tumblr_client Gem to export some posts to Tumblr. I'm using sequel but with a bit of tweking you could use this in a rails app.

export_to_tumblr.rake

require './sequel_config'
require './tumblr_config'

desc 'Export posts to Tumblr'
task :export_to_tumblr do
  client = Tumblr::Client.new
  Post.order(:published_at).each do |p|
    begin
      resp = client.text('some_tumblog',{
        title: p.title,
        body: p.body,
        date: p.published_at,
        format: 'markdown'
      })
      if resp['id']
        puts "Exported post #{p.title}: #{resp['id']}"
      else
        puts "Failed export post #{p.title}: #{resp}"
      end
    rescue => e
      puts "Error for post #{p.title}: #{e}"
    end
  end
end