Did John Tyler die in office?

No, John Tyler did not die in office. He served out a full term as president from 1841 to 1845. After leaving office, he would live another seventeen years before passing away at the age of seventy-one.
Although Tyler may not have died in office, he became president due to the death of his predecessor, William Henry Harrison. Tyler had been elected in 1840 as Harrison's Vice President, so when Harrison passed away after just thirty-one days in office, Tyler became president under the provisions of Article II, Section I, Clause VI of the United States Constitution, since superseded by the twenty-fifth Amendment.
Tyler may not have distinguished himself in the White House, but at least he's gone down in history as the first man to accede to the presidency on the death of the incumbent.