Did you know that players get burnt out too? I’m sure you did.
DMs get burnt out because they have a lot more riding on them than any individual player, but players have a uniquely-challenging task that the DM is insulated from - the player must invest in the game by playing (likely) only one character for a solid length of time. Whereas the DM gets to hop between characters or play the world with their descriptions thus breaking any monotony, the player cannot do this. Thing is, most players are simply happy to show up to a game, roll some dice, talk like they think their character would talk, whatever - most players are very easily satisfied.
However, sometimes you the player think you've messed up at maintaining the integrity of your character concept (roleplaying the "wrong" way), or you think you made the wrong decisions during character creation, or you simply become uninspired by your character and wish not to play them anymore. None of these issues are particularly problematic in the sense that worse comes to worst, you can roll up a new character - most DMs are very accommodating on this front. There is a value though to sticking it out with your character especially if the only reason you want to abandon them is because you think you messed up at roleplaying, made a decision you think is incongruent with your character concept, or, and this is a big one, did something that makes you feel like your character is pathetic.
If you the player become burnt out on the game itself, whether because it is being poorly managed by the DM, you realize you dislike the system, the game is boring, or for whatever other reason, there probably isn’t much you can do about it. Ideally you spoke with the DM a few times beforehand to try and iron out the things that led up to your burnout, and if nothing changed, you can simply leave the game. You ought to pull the DM aside after the session and succinctly explain the specific dynamics you link to your burnout. The reason I recommend doing so after the session is because if you do leave the game, that gives the DM the most time possible to adjust for your absence and further it is best to rip the band-aid off quickly. Thus, I advise that if you realize you’re burnt out of a game in between sessions, you still go to the next one, put in your all, and then let the DM know thereafter. However, it is better to get out of a bad situation quickly than smoothly if it comes down to it.
Typically, the above isn’t the case. Players rarely get burnt out on whole games, especially if they were serious about them to begin with. Typically you the player will feel burnout around your character. Often times, this type of burnout is a defensive mechanism against having to deal with narratively-difficult in-game situations, often times when you feel your character is a “damaged good”. Better to cast them aside and roll up another, you believe. For example, if you sustain an injury or curse that suddenly warps your character around a wholly-different narrative than you expected, you may want to cut them loose. Alternatively, it has to do with a disinterest stemming from the character concept itself; perhaps you made a fighting man and you actually want to play a magic-user, or you made a tragic-backstory hero and you actually want to play the schlub-backstory hero. You made a decision early on about your character that you now regret.
Let me make something clear here. When you make a character, I think you have some obligation to them, to see them through their journey, and that’s why it is so important to follow the right structure when designing your character concept. It is no small thing to give up a character you’ve spent a lot of time investing in, obviously. So the ideal answer to burnout around your character is to stick with them, pure and simple. Keep playing them and react to situations you find yourself in. But how?
If you feel constrained or suffocated by some earlier choice you made during the campaign or from some event, recognize that you probably are playing a specific response of many to that choice or event. For example, if your character’s parents died early on, you might be playing a brooding, emotionless character, but that is one potential response - plenty of people whose parents have died in the real world, even prematurely, aren’t brooding and emotionless. So if you are burnt out on playing the brooding, emotionless type, then give yourself permission to move on from that. You can incrementally, or suddenly (verisimilitude matters less than your investment in your character), shift your character’s behavior and general operating procedures. It often seems like you are locked in to a character concept, and to a large extent you are, but most players have an extremely-inaccurate view of the scope of their character concept - in the above example, all that you are locked into is having some sort of reaction to the fact that your parents died, which may or may not even come out during general roleplay interactions. How many things have happened in your character’s life? Probably a lot. How many of them do you know about? Extraordinarily little, even if you write the longest backstory you could (a really bad thing to do by the way). Act as if anything you do with your character is justified narratively because it probably is, at the very least, by something in your character’s long life before the campaign started. When you “transgress” the bounds which you assumed you had to operate within based upon your character concept, you discover something new. Giving yourself permission to discover rather than just to express is probably the best way to avoid, and to rid yourself of, burnout around your character, and to boot it vastly increases the amount of character growth you can experience throughout the campaign.
Worse comes to worst, you simply must retire your character and roll up a new one. This is okay, but you ought to figure out what you did wrong the first time around. One very likely culprit is that you pigeon-holed yourself too much during character creation when you forged your character concept. In fact if you focus on the wrong things during character creation, it can be very difficult to forge a long-lasting bond with your character during play - see How to Forge Characters for a guide on what to think about when you make a character.