I'm using OS X Yosemite 10.10.5. When using the default /bin/bash as your login shell, setting the following in .profile:
PROMPT_COMMAND='echo -ne "\033]0;${PWD/$HOME/~}\007"'
sets the Terminal tab title to ~ when you are in your home directory, or ~/sub/dir when in subdirectories.
However, Yosemite ships with bash version 3, and I prefer version 4, so I added /opt/local/bin/bash (which is version 4.3.42) to /etc/shells and used the chsh command to set my login shell to the MacPorts version. Everything else works fine, but when in my home directory, the tab title displays /Users/mattdmo instead of ~.
Did the PROMPT_COMMAND syntax change with version 4? How do I get the functionality I'm looking for? On my Linux (Ubuntu 15.04) VM, I'm using bash 4 and the same command works fine...