Centos 6 requires GLIBC 2.12 however a lot of new programs you would want to compile may need a newer glibc. You can't remove the old glibc since the whole OS is based on it but you can install the updated glibc alongside it and do an export pointing to your updated GLIBC.
mkdir ~/glibc_install; cd ~/glibc_install
tar zxvf glibc-2.14.tar.gz
sudo make install
The export should be done in ~/.bashrc or /etc/profile to make it permanent otherwise programs will have unpredictable results. If you are calling a remote/cron script that needs this GLIBC you should probably do the export in the script itself to be sure.