WASHINGTON—Vice President Mike Pence has filed three tax returns since taking office, but he hasn’t publicly released any of them.

As Democrats fight with President Trump about his unreleased tax returns amid a key deadline Tuesday, the vice president’s decision to keep his own recent tax documents under wraps has gotten far less attention. It is just as much of a break from his predecessors: Going back to Walter Mondale in the 1970s, all have disclosed their returns. Mr. Pence released 10 years of returns through 2015 during...