So I know that blind creatures/PCs/NPCs have disadvantage or immediate failure on saves and checks that require sight (unless they have blindsight or something like that) and that deaf creatures/PCs/NPCs have disadvantage or immediate failure on saves and checks that require hearing, but would a mute character have any disadvantages? I would think if they did it would most likely apply to something like persuasion, but I don't know for sure. If any of ya'll know that would be appreciated.

There are no saves that require a vocal component. But the character would not be able to yell "Look out!" to warn others of a danger (s)he perceived. 

Any communication would be at a disadvantage unless the characters used the same sign language or some other form of non-verbal communication.

And without Subtle Spell any spellcasting that requires verbal components is going to be impossible. 

Thanks for that info! I never would've known otherwise, haha

There's a misunderstanding that mute or non-verbal people are incapable of making any sound.  I wouldn't rule such a character would be incapable of sounding the alarm.  Maybe more nuanced and technical forms of communication may be disadvantaged.  Anyway, there was a discussion not too long back of a non-verbal bard and it seemed like "non-verbal people can't be spell casters because of the V component" has ready workarounds:

https://www.dndbeyond.com/forums/d-d-beyond-general/general-discussion/193294-a-mute-bard

Quote from MidnightPlat >>

There's a misunderstanding that mute or non-verbal people are incapable of making any sound.

The question wasn't "What are the workarounds" it was "What are the disadvantages". And I didn't state that the character couldn't "sound the alarm" or "are incapable of making any sound". I posted that they couldn't yell "Look out!" because that's the definition of being mute. 

The V in spellcasting stands for Verbal. Being mute means you're non-verbal. RAW the work-around is having the Subtle Spell ability. I'm not one to tell you that your fun is wrong. But if you're doing "workarounds" to make mute characters that can speak, blind characters that can see, etc., then (to me) it defeats the purpose. I want to play a character who was malnourished as a child so has a 4 in strength. But then he found Gauntlets of Ogre Power so he could live his dream of being a fighter. Ok, well if you enjoy that extra step, you do you. 

Quote from MidnightPlat >>

There's a misunderstanding that mute or non-verbal people are incapable of making any sound.  I wouldn't rule such a character would be incapable of sounding the alarm.  Maybe more nuanced and technical forms of communication may be disadvantaged.  Anyway, there was a discussion not too long back of a non-verbal bard and it seemed like "non-verbal people can't be spell casters because of the V component" has ready workarounds:

https://www.dndbeyond.com/forums/d-d-beyond-general/general-discussion/193294-a-mute-bard

This is one interpretation used by people who want to play the "mute caster" bit, but it's not strictly accurate to the RAW as printed. The first line of Vocal Components in RAW is "Most spells require the chanting of mystic words." Strictly speaking, the PHB establishes that you need to be able to verbalize phonemes to cast a spell; you can't just grunt or whistle. Notably, there's a creature block for clerics of a particular deity from the FR Orc pantheon (a Hand of Yurtrus) who have their tongues removed, and their spellcasting feature specifically notes that they cast without V components. With both the exact text from the PHB and a specific example of a creature who fits the conventional definition of "mute" needing specific language to get around V components, it seems clear that it's both RAW and RAI that you need to be capable of speech to meet the requirements for a V component. I'm not saying this isn't something that a table can't set aside if the DM is game for accommodating your model, but it's not actually in line with the material.

I think it would be fun for about 5 minutes of play time. And I'm talking as a player in the group with the mute character or as the GM.

As a GM I would just say no.