How to allow user to preserve environment with sudo?

You can use the SETENV "Tag" in your sudoers file, as in :

deploy  ALL=(ALL)       SETENV: /usr/bin/git, /etc/init.d/httpd*, /sbin/service, /usr/bin/make, /bin/echo

Or, to combine it with NOPASSWD:

deploy  ALL=(ALL)       NOPASSWD:SETENV: /usr/bin/git, /etc/init.d/httpd*, /sbin/service, /usr/bin/make, /bin/echo

Relevant excerpt from the sudoers man :

SETENV and NOSETENV

These tags override the value of the setenv option on a per-command basis. Note that if SETENV has been set for a command, the user may disable the env_reset option from the command line via the -E option. Additionally, environment variables set on the command line are not subject to the restrictions imposed by env_check, env_delete, or env_keep. As such, only trusted users should be allowed to set variables in this manner. If the command matched is ALL, the SETENV tag is implied for that command; this default may be overridden by use of the NOSETENV tag.

Don’t specify the -E option. Using -E you´re saying that all the environment variables for the user deploy should be preserved, not only GIT_DIR

Running sudo echo $GIT_DIR should work because you've added GIT_DIR to the env_keep list