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We consider the effects of interface bound states on the electrical shot noise in tunnel junctions formed between normal metals and one-dimensional (1D) or two-dimensional (2D) Rashba semiconductors with proximity-induced s-wave pairing potential. We investigate how the shot noise properties vary as the interface bound state is evolved from a non-zero energy bound state to a zero-energy bound state. We show that in both 1D and 2D tunnel junctions, the ratio of the noise power to the charge current in the vicinity of zero bias voltage may be enhanced significantly due to the induction of the midgap interface bound state. But as the interface bound state evolves from a non-zero energy bound state to a zero-energy bound state, this ratio tends to vanish completely at zero bias voltage in 1D tunnel junctions, while in 2D tunnel junctions it decreases smoothly to the usual classical Schottky value for the normal state. Some other important aspects of the shot noise properties in such tunnel junctions are also clarified.