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I don't think this is a Lithuanian name, but Lithuania was part of the Russian Empire, and I think there is still an ethnic Russian community there. Lithuanian surnames seem, in my limited experience, to need an S at the end, though when borne by females they have different endings. Think of Vitas Gerulaitis, that's a very Lithuanian name. Many Lithuanians have surnames that are Polish in origin, but they adapt them. So Zamecki becomes Zameckis, Marcinkiewicz becomes Marcincovicius.
With regard to an Italian surname Putin, I think that's possible. It would probably be Venetian or Piedmontese. There are some good Italian surname websites if you know some Italian. Try www.comune.occhiobello.ro.it/cognomi. Even if you don't know any Italian just finding the name Putin there will prove your point.
Putin as a French surname? It doesn't feature in Albert Dauzaat's dictionary of French surnames, though that in itself does not settle the argument.