I'd like to create a language with only consonants, using the letters b, ch, d, f, g(hard), j, k, l, m, n, p, r, s, sh, t, th (voiced and unvoiced), v, z, and (most importantly) h. I feel like the h would just become a mixture of unvoiced vowels as humans spoke the language. Is this likely to happen over time?
Well, over time, you're almost certainly going to gain vowels. Vowels are very easy to pronounce (think about the first sounds babies make as they're learning to use their mouths), and very easy to hear. So the evolutionary forces that apply to any language in productive use will tend to introduce vowels. They're just so useful!
If you want this language to be naturally spoken for a long time without evolving vowels, you'll need some way to prevent that. One way would be to have the speakers lack vocal cords, or not be able to use them (so they're always whispering). This makes vowels a lot harder to hear, so they lose some of their evolutionary advantage. But this will probably remove all the nasals, too, since those are practically impossible to hear without voicing.
I personally don't think it is possible to have a language without vowels unless you are talking about a sign language or some code like language that doesn't use syllables (as in clicks or taps or dots and dashes like morse code). Though I think @Draconis has a point, maybe if they whispered a lot, or maybe if the speakers were snake people and hissed everything and replaced the vowels with 'S's.
However far more achievable would be to have little to no supporting and elongating vowels, and as little vowels as possible.
Nothing would be drawn out, all sounds immediately followed by another sound when two or more phonemes are put together. Very up and down sounding, like stuttering on purpose but language.
Instead of having vowels stressing/not stressing/raising them up or down for the sake of tone, I theorize that rhythm would instead become far more crucial. Something like iambic pentameter must exist especially for more formal kinds of talk/occasions. And people singing or just arguing probably always sounds like rapping/a rap battle.