David Cook, Sonja Mapes and I have a package for Macaulay 2 which draws pictures of posets, either with or without node labels. It will produce the TikZ code to include these in papers as well.
How large though is "very large"? While there are a number of built-in enumerators in our package (lcm lattices, hyperplane arrangement lattices, noncrossing partition lattices, etc.) which can produce fairly large posets quickly, the only limitation might be how you're storing or inputting these posets.