Mac - Convert from Case-Sensitive to Case-Insensitive File System
It looks like you can restore from time machine to a case-insensitive file system from a case-sensitive one. See Apple Forums here.
I would highly recommend that you perform a total system backup using Carbon Copy Cloner.
Carbon Copy Cloner does support restoring to a case-insensitive system from a case-sensitive one - see this.
In fact, I would recommend CCC over TM, but you should use what you feel comfortable with. I would also boot to the CCC image and make sure all my stuff was there before I wiped my internal drive and did the restore. CCC makes a bootable back up (which is another reason it's awesome).
I haven't used it, but iPartition claims to be able to do nondestructive conversion between case-sensitive and case-insensitive HFS+.
Supposing the program you're using is just insisting on having a case-insensitive FS while your data is on a case-sensitive FS, you can sometimes get away with just mirroring the directory structure over to the case-insensitive FS, and use symbolic links for the files. This way you avoid duplicating potentially large or numerous files.
The "lndir" command from the X11 distribution for OSX does just that (in Mountain Lion this is in Xquartz, which you can get from http://xquartz.macosforge.org/landing/).
For instance, consider you want to clone Foo.app (which resides in a case-sensitive FS) to a case-insensitive FS, which we assume here to be the root filesystem, you might do something like:
$ cd $FOO_APP_BASE_DIR # Foo.app's parent directory
$ lndir $PWD/Foo.app /Applications/Foo.app # clone it into /Applications
You may now open /Applications/Foo.app and it won't complain about case-sensitivity.