I usually type TV characters based on how much they remind me of the subjective type "archetypes" that I have in my head and apply logic afterwards to confirm or deny my guesses. So apologies if my logic is shaky.
Concur that Ted is an ISFJ. He seems rooted in his expectations, as what his inf-Ne has observed in the past will be repeated as his dom-Si anticipates. His traditionalist (as I see it, the girl is either "the one" or she isn't. he's classic and modernistic) approach to dating is unsullied by his past experiences and failures, the strength of his Si is able to hold onto such notions. To apply a socionics concept, Si doms have "role Ni", my underdeveloped understanding of it is they disguise themselves in casual situations by taking on a superficial role- when he flirts he puts on a front of sophistication, the master of the arts and concepts, he wants to be seen as being from a higher level of understanding from others but when he's acting sincerely he doesn't do this at all. His aux-Fe is hard to argue, it's just there to me. Although he is somewhat self absorbed he is socially attuned at all times, he's aware of the emotions of people around him and seems to regard as part of the equation as he guides group decisions and even personal decisions to a lesser extent.
Marshal is INFP I think. He is absorbed intimately in mutual happiness with Lily, he is completely self aware of this as well and uses the self awareness to further the deepen the depth of his love for her. He doesn't have an Fe agenda, nor does he have the detachable quality that Fe has (where emotional expressions are kept in a toolbox and used to suit the situation). As for Ne, he is creative in his romance and resolves conflicts by seeking an entirely new angle to express himself from. He is enthusiastic about new concepts introduced in his friend group.
Lily is definitely tougher. I had thought ENFJ but honestly I have no argument at the moment. I'm not sure if I even agree with myself anymore.
Barney is an ENFP
Robin is ISTP
g2g I might flesh these out later