No. Despite having a diplomatic passport, his hosts push him and his entourage past all of that because every visit he makes is a state visit. The passport process is handled by a State Department employee for the entire group. Everything is planned ahead of time with the host country. Plans are prepared for his return in a similar manner.
When visiting a foreign nation, a president and his entourage essentially have everything arranged for them. A head of state and probably some top officials with him should not have to go through immigration processes in a foreign nation; nevertheless, a few employees in the middle and lower echelons of the hierarchy might. The passport and other documentation of the head of state are typically not even requested by the majority of hosting nations.
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Hope this answers this question.
Others can reply to this thread with their own answers.