The answer depends upon whether you are referring to the tubes themselves or an entire tube-based circuit.
First, lets clarify that the popular term burn-in really refers to forming dielectrics. This is where electrons find and groove a path of least resistance through the insulation materials surrounding the conductors. Dielectrics (teflon, polyethylene, glass epoxy boards, etc...) act as an impediments or road blocks to the free flow of a signal.
The actual vacuum tubes themselves require little or no burn-in because they have so little dielectric material to form in the first place. The conducting elements are surrounded primarily by a nearly perfect vacuum, which is a superior dielectric to all others including air.
Tube circuits like all circuits which contain capacitors, resistors, wire and circuit traces, require significant burn-in because all of those passive parts contain all kinds of plastic dielectrics which need to form to sound best.
Transistor amplifiers require and benefit from even more burn-in because unlike vacuum tubes, the transistors themselves contain significant dielectric material which also needs to be formed.
If there is any advantage for tubes over transistors in relation to the idea of burn-in, it is that the transistor contains lots of solid dielectric material tightly packed around the conducting elements, where a vacuum tube contains virtually nothing.