I'd like to make a language that only has consonants. Since I kind of feel like the letter "h" is a consonant, I'd prefer to omit it. The language would include b, ch, d, f, g(hard), j, k, l, m, n, ng, p, r, s, t, th(voiced and unvoiced), v, and z. I think maybe humans wouldn't be able to speak it without also pronouncing h and/or vowels. My question is, what (if any) problems in pronunciation would arise if humans tried to speak a language like this?
The IPA draws a sharp distinction between "vowels" and "consonants", but really, there's nothing you can measure in a spectrogram to decide if something is a vowel or not.
[u] are basically identical, in terms of objective acoustics. The real difference is phonological.
In other words, the true difference between a vowel and a consonant is how it's used in a language. "Vowels" act as the core of a syllable, while "consonants" attach to either side. In my variety of English,
/r/ can be used as syllable cores (the second syllables of "rhythm", "button", "bottle", and "dollar"), which in some sense makes them "vowels". (The most common term for this is "syllabic consonants".) For me, "mirror" actually has
/r/ acting as both a consonant and a vowel in the same syllable!
So if you want to make a language that doesn't use any of the letters the IPA deems to be vowels, that's totally doable. If you don't want to have any syllable cores, that gets harder. You need to decide how you're sticking your phonemes together in order to avoid having anything that could be considered a syllable or its core.
To finish with a conlang example, I once made a very limited conlang for a tabletop game, in which a character was named Gzhzaqrh:
/gʒ̩.ˈzɑ.qʁ̩/. In this name, the zh and rh act as syllable cores, even though they're normally considered consonants. The result tends to sound strange and unusual to humans, which is probably what you're going for. (In the game, it was spoken by orcs.) But the players didn't have much difficulty pronouncing the name, once they got used to it (though as English-speakers they usually substituted velars for uvulars).
There is no problem in an all-consonant language as natural languages like Nuxalk aka Bella Coola demonstrate. The example language actually has vowels, but it is famous for its long vowelfree consonant sequences and even full sentences without vowels.
Your choice of consonants including a lot of voiceless fricatives may be problematic because the different fricatives can easily be mistaken for each other in a noisy environment, when transmitted over a traditional phone line, or when the listener is hearing impaired. But again, Nuxalk also goes with a range of voiceless fricatives, so this is probably also not a problem.