tcz files are packages for tinycore, tce is the package manager system...... similar to .deb files in debian retrieved with the package manager apt.
Lazarus itself is not available on tinycore....not sure how easy/hard it would be to add. Does it compile to an binary executable? What are the library dependencies of the file? tinycore is binary compatible with programs compiled on wheezy.
You can install any file you want on the system, but without doing a backup, it only resides in ram, and will be gone after a reboot. Just copy the application to your home directory, then do a backup....there is a backup link on the desktop. (Assuming you are running the X window image of tinycore.)