If I am quoting text messages where the person continuously uses abbreviations such as u instead of you or ur instead of your, when I quote these am I supposed to put [sic] by them to show I am not spelling them incorrectly? Or do the quotation marks already show that they are written exactly as the original? Thanks so much!
If it is clear you are quoting text messages, your audience will (should) expect appropriate spelling for text messages. That means that u, ur, 2B, l8 and such things are not considered misspellings but rather domain-specific dialect.
Adding [sic] after every such acronym/spelling not only makes the quoted message (even more) unreadable, you may also give your audience the idea that you have absolutely no idea how text messages are spelled nowadays... or at least, they may think you pedantic.
Now, if there is an actual misspelling, you can use [sic] to indicate that, but generally the effect of judging every texting-spelling as “wrong” would look bad:
“Will B[sic] @[sic] party 20:30. CU[sic] L8er[sic]!”