For real animal noises, crocodiles and aligators make drum thump noises when their mouths snap shut. Humans can make drum thump noises but only as vowels, so we couldn't vocalize another sound over it. Rodents grind/chatter their teeth (bruxing); humans can do this but we are too big to do it at the same frequency, and it is injurious to us to do it for long.
I suspect that scaling humans up or down is an easier way to make the sounds you're asking about than looking for animals. If whistles count as consonants and unproducible notes count as unproducible sounds, it would be easy to imagine a mouth configuration that could produce a lower or higher register than humans can duplicate, just by scaling the relevant parts up or down, exactly as scaling up a flute makes flute sounds that a smaller flute cannot duplicate. Unvoiced tongue and lip trills would likewise be easy to scale up or down, creating trill rates lower or higher than humans can duplicate. Since humans come in a variety of shapes and sizes, with plenty of humans in any extreme, there are plenty of humans that can make lip/tongue/teeth based consonant sounds that most other humans are physiologically incapable of making, simply because they have different mouth shapes.
If you allowed real structures other than real animal lips, tongues, and teeth, but still excluding throat configurations, it's easy to imagine something like a rattlesnake tail as a mouth part, or even a person with little jingle bells as lip piercings, who could easily create jingly consonant sounds that no unmodified human could ever duplicate.
If you allow for inability to produce certain sounds without extensive training, there are many human languages that other humans cannot duplicate. For an extreme example, see: Taa, which is almost impossible for adult learners to reproduce - but less difficult examples are everywhere. Think of the Korean consonant b/p consonant bieup, which most English-speakers can't produce, or the English z, which most Korean speakers can't produce. An alien with the perfectly human name Zachary Hill who encountered Korean astronauts might reasonably conclude that his name was entirely "unpronounceable in your human tongue". An alien with the perfectly human name ǃnˤù.ṵ would find the same for almost any variety of human astronaut, unless the Taa beat the rest of us to first contact.