Feynman diagrams with classical apparatus on the perturbative region In QFT, one usually simplifies the interaction between fields and classical apparatus (sources, detectors, etc.) by assuming the classical devices only interact with the asymptotic on-shell states
I'm interested in physical scenarios where such approximation would give incorrect results.
Conceivably one can put a logical or measurement device (which we can simplify as a bit or qubit state in some system) with enough coupling and/or proximity to a source such that one cannot assume that the device bit state must interact only via asymptotic states, in the computation of the Feynman diagram amplitudes
 A: For QED this corresponds to bound states, and the solution is well known, if one is willing to admit that bound state calculations do not rely upon the assumption of detectors interacting only asymptotically.  For example, some of the tricks employed involve substituting the asymptotically free electron states with electron states in a known external classical field W. H. Furry, 1951. This is essentially the calculational consequence of the philosophical statement of the separation between field interactions and classical apparati. Said another way, what does it mean to get close to a piece of electronic equipment? Well, it means to be in the constant presence of an external (essentially pervasive from the quantum perspective) EM field. There is a review of modern physics by Bodwin, Yennie and Gregorio 1985 that reviews various bound state calculations in QED. It is emphasized there that the calculations are nonperturbative. One way this affects the calculations is that one cannot simply count powers of alpha to get estimates for perturbative corrections obviously owing to the nonperturbative nature of the system. The approximations used are very different too. In general, one arrives at a very different equation of motion with an inherently different propagator, i.e. the Bethe-Salpeter equation.
Also, such a circumstance can occur in a standard condensed matter system that is not far away from its boundary. To make this more concrete, think of lattice periodicity that is spatially spontaneously broken at an interface. One can handle this in terms of standard techniques for spontaneously broken symmetries, in this example the lattice symmetry group. But the states at the boundary/interface are interacting, and are not asymptotically free.
So I would summarize the answer to your question (though there was no question mark, only a statement of curiosity) as follows. If one wishes to incorporate device interactions then one must choose the interaction one has in mind and then analyze the QFT consequences through this new lens. I say the answer is 'known' only because such ideas of viewing QFT through an interacting lens, or as Furry puts it, a mix of the free and interacting pictures has been done before. Has it been done for all possible physically interesting configurations? Definitively, no.
A: The question seems to have two interconnected parts.

In QFT, one usually simplifies the interaction between fields and classical apparatus (sources, detectors, etc.) by assuming the classical devices only interact with the asymptotic on-shell states
I'm interested in physical scenarios where such approximation would give incorrect results.

One could read the question here as "Are there situations where quantum field properties beyond asymptotics/scattering are measured?". Indeed, this is frequently done in ultra-cold quantum gases and other quantum simulator platforms (see, for example, Phys. Rev. Lett. 105, 190403 (2010) or  New J. Phys. 19, 023030 (2017)). In these experiments, the full dynamics of the quantum field can be observed. The field itself is given by the Bose-Einstein condensate. While this is a non-relativistic situation, quantum field properties certainly play a role in these systems, and the papers I provided also show that relativistic scenarios can be simulated.

Conceivably one can put a logical or measurement device (which we can simplify as a bit or qubit state in some system) with enough coupling and/or proximity to a source such that one cannot assume that the device bit state must interact only via asymptotic states, in the computation of the Feynman diagram amplitudes

Arguably, this is also done in ultra-cold quantum gases, where one can image individual atoms. While the imaging process itself could potentially be regarded as an asymptotic process, the measurement is really made on the atom. Since the atomic position can be the dynamical property of interest, this constitutes such a 'local'/non-asymptotic measurement of a quantum field.
