When an audio recording has lower quality, it means two things: some information is lost, and some noise is added.
Obviously if enough information is lost, there won't be enough left to represent the sounds of the language anymore, so the question is really a matter of degree. But the important question is where do you start having problems.
As information is lost, it becomes harder to distinguish sounds. Actually, the auditory differences between two different sounds can be very subtle; humans who acquire a language are able to distinguish between two phonemes that really are very similar to each other from an acoustical perspective.
This means that a low quality recording is potentially lacking in important phonemic information.
This may mean that the learner is unable to notice the difference between two similar phonemes in the language, or that they are unable to distinguish between a sound in their own language and the language they are learning. For example, if they are learning Japanese, they may not distinguish between the English "r" and the Japanese "r" (actually a flap [ɾ]). So they may just pronounce the sound as a sound in their own language, and this pronunciation could become fossilized.
Of course, this will be most likely to happen if the learner is mostly learning pronunciation through one source, the cassette tapes. My recommendation would be never to learn the language through the one source. If at all possible, include some real interaction with native speakers in your language learning routine. Then you'll be getting rich phonological input, and the interaction will provide an opportunity to get feedback about your pronunciation too.