In March 1992 Burundi adopted a new constitution. It was promulgated on 13 March of that year, following a referendum four days earlier in which over 90% of those voting approved the new text.
Art. 61 stated that the president is directly elected for a five-year once-renewable term.
Art. 72 of the 1992 text stated that the president named the prime minister and ended his functions (i.e. a direct copy of the current French constitution. In fact, much of the constitution was a direct copy of the French one).
Art. 86 stated that the prime minister was part of the government.
Art. 89 said that the prime minister was responsible before the president and the National Assembly. So, Burundi was an example of a president-parliamentary form of semi-presidentialism.
Art. 134 stated that the government could ask for a vote of confidence in the Assembly. If defeated, by an absolute majority of votes, the government had to resign.
Art. 139 stated that the National Assembly could lodge a motion of censure. Again, if passed, with a three-fifths majority this time, the government had to resign.
As far as I understand it, the 1992 constitution was amended in January 1994. This followed a coup in October 1993 when the Tutsi-led army killed the Hutu president, Melchior Ndadaye, who had been elected in a pretty free and fair election just a few months earlier. The January 1994 constitutional amendment stated that the president would be elected by the National Assembly. I do not have the text of the amendment, but, to my knowledge, Burundi ceases to be semi-presidential at this point.
The constitution was officially suspended following the coup in July 1996.
There is a very informative article on the 1993 elections by Filip Reyntjens in The Journal of Modern African Studies, Vol. 31, No. 4. (Dec., 1993), pp. 563-583.