I am not aware of any evidence for violation of energy or momentum conservation - they are at heart of Lagrangian mechanics we successfully use from QFT to GRT.
Regarding virtual particles - they are used in Feynman diagrams in perturbative approximation of QFT - assuming QFT is fundamental, perturbative QFT/Feynman diagrams is still an effective picture - practical approximation ... leading to countless number of divergences, usually removed by hand.
But it is extremely universal practical tool defining objects as point particles through their interactions - it is very general algebra on particle-like objects ... like algebra properly concluding that "apple + apple = two apples" without any insights what apple is ... which can also handle non-point objects like fields by approximating them with a series of virtual particles.
The basic example is Coulomb interaction e.g. proton - electron, which in pertubative (approximation of) QFT is handled with a series of point-like photons, instead of continuous EM field.
It brought dangerous common misconception that EM field is always quantized, while it is just a continuous field, which optical photons are quantized due to discrete atomic energy levels ... but e.g. linear antenna produces cylindrically symmetric EM radiation - which energy density drops like 1/r to 0 - cannot be quantized to individual discrete photons localizing finite portions of energy.
We have lots of quasi-paricles especially in solid state physics - starting with phonons: classically just Fourier/normal modes of the lattices, but perturbatve QFT treats them as real particles ... point-like.
There is also virtual pair creation - while we imagine pair creation as a zero-one process, it is in fact continuous - field can perform a tiny step toward pair creation, represented as real (virtual) pair creation in perturbative QFT. Continuity of this process is nicely seen using topological charge as charge: