Where is this coming from? He never implied or gave any idea that he was preventing her from doing anything? All he ever did was protect her when shit went south with Joffrey. No it wasn't his responsibility at all. By this lovely logic, I should blame my Prime Minister for literally every issues in my city, or mayor because everything revolves to their responsibility. Why? Because reasons.
Tyrion never had her captive nor was it his duty or responsibility to help her.. just like it wasn't his responsibility to help any other random person. His duty was taking care of politics in the city and trying to contain Joffrey as semi-sane which for the most part, he did.
If you are being held by the federal government as a high-value prisoner, then yes, it does come down to your prime minister.
Sansa was a prisoner of the Lannisters. Tyrion, as the head Lannister in charge of KL, was responsible for keeping her prisoner (note that, in S2E1, he points out that Cersei was supposed to capture both Sansa and Arya, and with Arya missing, Tywin will be angry with her).
Thought experiment: Varys brings Tyrion word that Sansa is planning to escape. Does he:
(a) do nothing and let her leave.
(b) stop her from leaving because she's a Lannister prisoner.
Hint: the correct answer is (b).