Can I replace the TINI OS with my own monitor program?
Yes. TINI is built around the DS80C390 CPU, which is an 8051-compatible chip. So you can program it with 8051 assembly or in C using a compiler that can generate 8051 machine code. Then convert your program into Intel Hex format and download it into bank 0 of the Flash ROM and viola, you have your own OS. To take full advantage of TINI, you will need a TCP/IP stack, serial driver, etc. etc. And be aware that doing this will overwrite the bootloader code normally stored in bank 0, so take care.
Karl Bongers has done this and addressed some of these issues on his web page at: http://www.turbobit.com/tini.htm