You're going to need the CO2 tank.
First, when you first keg the beer, you'll need to "purge" the oxygen from the headspace of the keg and replace it with CO2 to prevent your beer from oxidizing.
Secondly, you'll need to pressurize the headspace in the keg to ensure the lid holds a firm seal until the yeast produce enough of their own CO2 to carbonate the beer. If you seal the keg up with no pressure initially, the yeast will go to work on the priming sugar, and will start to produce CO2. However, initially the pressure will be very low; so low that it might leak out the seals of the keg if there are any gaps, never actually carbonating the beer. By pressurizing the keg initially with some CO2 from a tank, this "seats" the lid and seals and ensures that any CO2 produced by the yeast consuming the priming sugar remains in the beer, rather than venting out through poorly-seated seals.
Finally, you'll need CO2 to push the beer out the serving lines. You could use a hand pump, but you'd be introducing oxygen into your keg and would oxidize your beer over time. If you plan on drinking the entire keg in one night (like for a party), then that might be good enough. But there's really no good substitute for using CO2 to serve your beer. If cost is an issue, you can use a smaller, paintball-gun-style CO2 tank for serving the beer.