Investigating the porosity–permeability evolution in carbonate rocks during long-term infiltration is a challenging hydrodynamic problem in rock engineering (hydroelectric engineering, CO2 geological storage, and oil/gas development). In general, we consider the evolution of porosity and permeability to be positively correlated. However, the complex pore structures in carbonate rocks include multiscale pore-fracture structures, such as intergranular pores, dissolution pores, and even fractures. Although the permeability levels of carbonate rocks are similar, the evolution of the pore structure is highly variable. Here, we use the “rock seepage coupled nuclear magnetic resonance (NMR) online analysis and imaging system” platform to conduct a long-term penetration test in carbonate rocks under the coupling of stress and reactive fluid. The variation in the multiscale pore volume was quantitatively characterized by the inversion of the T2 spectrum. Subsequently, in combination with permeability measurement, the permeability-porosity evolution during long-term infiltration can be roughly divided into three stages. The experimental results show that the evolution of permeability exhibits noticeable stress sensitivity at the initial stage of the test (0–156 h). Since the compression of the pore volume is the dominant mechanism, the permeability decreases with porosity at this stage. A negative correlation between the evolution of porosity and permeability was observed during the subsequent stage of the test (156–400 h). This phenomenon may be attributed to mineral dissolution at this stage, causing the cementation to be exfoliated from the pore surface and to re-precipitate at the pore throat, locally blocking the seepage channels. As the test continued (400–720 h), mineral dissolution transitioned from micropores to mesopores and macropores, with a synergistic increase in porosity and permeability. These results revealed the competing mechanism of stress compression and mineral dissolution during long-term infiltration, providing an accurate prediction of the seepage properties in carbonate rocks.