Updated for Capistrano 3.

deploy.rb

# encoding: UTF-8

# Place in config/deploy.rb

namespace :rails do
  desc "Open the rails console on each of the remote servers"
  task :console do
    on roles(:app) do |host| #does it for each host, bad.
      rails_env = fetch(:stage)
      execute_interactively "ruby #{current_path}/script/rails console #{rails_env}"  
    end
  end

  desc "Open the rails dbconsole on each of the remote servers"
  task :dbconsole do
    on roles(:db) do |host| #does it for each host, bad.
      rails_env = fetch(:stage)
      execute_interactively "ruby #{current_path}/script/rails dbconsole #{rails_env}"  
    end
  end

  def execute_interactively(command)
    user = fetch(:user)
    port = fetch(:port) || 22
    exec "ssh -l #{user} #{host} -p #{port} -t 'cd #{deploy_to}/current && #{command}'"
  end
end

For rbenv:

namespace :rails do
  desc "Open the rails console"
  task :console do
    on roles(:app) do
      rails_env = fetch(:rails_env, 'production')
      execute_interactively "$HOME/.rbenv/bin/rbenv exec bundle exec rails console #{rails_env}"
    end
  end

  desc "Open the rails dbconsole"
  task :dbconsole do
    on roles(:app) do
      rails_env = fetch(:rails_env, 'production')
      execute_interactively "$HOME/.rbenv/bin/rbenv exec bundle exec rails dbconsole #{rails_env}"
    end
  end

  def execute_interactively(command)
    user = fetch(:user)
    port = fetch(:port) || 22
    exec "ssh -l #{user} #{host} -p #{port} -t 'cd #{deploy_to}/current && #{command}'"
  end
end

In case anyone still have issues with rbenv and @DmytroLukianov solution, I tuned it a bit and it worked for me:

namespace :rails do
  desc 'Start a rails console'
  # Usage: cap production rails:console
  task :console do
    on roles(:app) do
      exec "ssh #{host.user}@#{host.hostname} -p #{host.port || 22} -t 'cd #{current_path} && $HOME/.rbenv/bin/rbenv exec bundle exec rails console'"
    end
  end
end