They are all very similar, they mean the same thing but with varying levels of extremity.
to be sick of...is extreme and implies something is making you sick and you can't continue, it's also quite insulting to say to someone "I'm sick of your behaviour/you". I would say this is the most harsh form, there's an undertone of disgust within the statement - however I generally use it a lot in my every day language because it carries with it a sense of finality or an ultimatum. For example, whatever it is is bothering you so much that it is making you sick, hence killing you. So you are expressing a feeling that either this thing needs to stop or you 'stop'. Often can even be used jokingly to over exagerate something that might be bothering you but is actually not that bad. (body language and tone are important here).
To be fed up with... The middle ground one, it's pretty stern, it expresses the same kind of feeling that something is bothering you but with less negativity. Something is bothering you but you can deal with it, it's not the end...that's probably the best way I can describe it. This would be something I would expect a teacher to say to a student who is being annoying on a consistent basis or something along those lines.
To be tired of...this can be either a soft form of expressing annoance, it's often said in a resigned manner or a pleading manner, it's less aggresive.
TLDR;
To be sick of = For fucks sake, I can't take it anymore!
To be fed up with = This is really annoying, I want it to stop.
To be tired of = I just can't be bothered dealing with this and i wish it would stop.
Like with any language, depending on your tone you can make "to be sick of" sound non serious and you can make to be tired of sound super serious and final but even when doing this, they aren't exactly the same in flavour and feeling.