When the tilde is quoted, it is a regular character and should not be expanded by the shell into your home directory. Therefore, the shell is behaving as expected for the above example.
The tilde here still isn't required to be expanded because the / following the ~ is quoted.

Note, however, that the sequences of commands:

Code:

MY_PATH=~/"bin/200_yast_install"
ls "$MY_PATH"

Code:

MY_PATH="$HOME/bin/200_yast_install"
ls "$MY_PATH"

should do what you want to do. (Note the use of double-quotes around the expansion of MY_PATH in the ls command. Without the quotes there, any spaces that had been quoted when you assigned the string to MY_PATH will be parsed by the shell into separate arguments for ls.

Expanding on what RudiC said...
The above rules apply whenever a ~ expansion is performed. In an assignment statement (and only in an assignment statement), any ~ up through and including the next following colon or slash, if there is one in the current word, must be unquoted for ~ expansion to occur. This allows multiple ~s to be expanded when setting the PATH, CDPATH and similar variables. Note also that the general format of a ~ expansion is any one of the three forms: ~logname, ~logname/, and ~logname: which will expand to the login directory for the user named logname on your system (followed by a colon or slash, respectively, in the last two forms). As you already know, if you omit logname in any of these forms the ~ expands to the quoted expansion of $HOME. ~ expansions only occur when ~ is the first character in a word except for the cases stated above in assignment statements.