In the comments, Constantin says that the question came from a student. I remember that when I was a student getting ready to study the Lebesgue integral for the first time, and discussing it with my peers and with other students who were slightly ahead of us, the motivation for the Lebesgue integral was not particularly clear. Someone pointed out that weird functions like the characteristic function of the rationals were Lebesgue integrable but not Riemann integrable. Someone else said that the Lebesgue theory allowed you to prove nicer theorems about interchanging the order of operations (i.e., limits, integrals, derivatives). For my peers who were more interested in physics than in math, these motivations were not convincing. I mean, who cares about the characteristic function of the rationals? And rigorous justification of the interchanging of limits is the paradigmatic example of something that fussy mathematicians sweat over but physicists just do. Does the budding physicist really need to learn all that math?

Implicitly, I think there are two separate questions being conflated here. The first question is the question being asked on the surface, namely whether rigorous mathematical theories of integration beyond the Riemann integral ever find application in physics. The second question, which is perhaps closer to the real concern of the student, is whether the conceptual understanding of integration afforded by undergraduate courses on the Riemann integral is good enough for most physicists (and engineers, perhaps).

Others have answered the first question well. I'd like to say a bit about the second question. I believe that there are some concepts that virtually every physicist/engineer is going to have to learn about, that go beyond what the "freshman calculus" treatment of integration gives you. Perhaps the first one that the student will encounter is the Dirac delta function. Now here's an interesting thing about the Dirac delta function: If you want to deal with it rigorously, then the standard course on Lebesgue integration may not help you very much. Instead, you either need to introduce the theory of distributions, or if you're content with a more lowbrow treatment that suffices for basic applications, you'll need the Riemann–Stieltjes integral. But perhaps more importantly, it's not clear to me that most physicists really "need" any mathematically rigorous treatment of the Dirac delta function. The mathematical physicists do, but do "regular users" need the rigor? Not clear.

I think the story is similar for the more advanced integration concepts that show up in physics. For quantum field theory, you absolutely need the Feynman path integral. But do the physicists need a mathematically rigorous treatment of the Feynman path integral to do their work? Clearly not, since the physicists have been merrily using the path integral non-rigorously for eons. Is taking a course on Lebesgue integration going to help the student master the Feynman path integral? Seems doubtful to me.

So I would answer as follows. Should budding physicists expect to have to expand their conceptual understanding of integration beyond the Riemann integral? Yes. Does this mean that they need to study more integration theory, in particular the Lebesgue integral, from a rigorous mathematical standpoint? The answer to this is less clear. Certain concepts from probability theory and functional analysis are indispensable in physics, and the Lebesgue theory forms the mathematical foundation for these subjects. So if you're the kind who benefits from being exposed to a rigorous treatment of the foundations, then you'll want to study Lebesgue integration. But many physicists get by with an intuitive understanding of those concepts without worrying too much about the mathematical niceties.