The normal case for TTY (as an abbreviation of "teletype" among others) would be all upper-case (see the Teleprinter page on Wikipedia for some confirmation and some history).
In fact, the use of all upper-case for such abbreviations (be they acronyms or -- as in this case -- initialisms) is very common1 (see Initialisms at Oxford Dictionaries). Plurals and possessives are then formed in the normal way ("serveral TTYs in a row" or "the TTY's screen was broken").
However, on Linux and other flavours of Unix, your current "teletype" (or however you connect) is mapped in the file-system to the (pseudo) file
/dev/tty. Because *nix filenames are case-sensitive, if you are explicitly referring to "the tty device" (meaning the pseudo-file, and possibly the device-driver code behind it) using lower-case would be more common.
1 I would have said all upper-case was near-universal (with the main exception being anything driven by marketing people looking to be "individual" or trendy). However, this Language Log entry at UPenn noted that there is (or was in 2008 when posted) a growing tendency in British newspapers to only use upper-case for the first letter. Not sure I've noticed this, but I try to avoid newspapers unless doing the crossword!