If you plan on doing a very simple community tank with danios and tetras, and maybe a few barbs, cories, you don't need a heater at all.
I've had my tropical fish survive temps as low as 68oF and be perfectly fine. My twenty gallon is still thriving after five years of no heater.
If you plan on more delicate fish, such as angels, discus, neons, livebearers, baby fry, and something that looks fancy, a heater might be a must. Some live plants also require more tropical temps for growth.
The trick is to keep the tank from changing temperatures too drastically. If a tank goes from say, 84oF to 76oF to 68oF over the course of a week, that's fine, the fish have time to acclimate to the decreasing temperature. If you were to go from 84oF to 68oF in a span of ten minutes, then yes, you can shock your fish to their death.
Non-heated tanks are likely to have only a 2o temp difference between night and day. This is fine.
Keep your tank away from windows or from A/C and heating vents. These things can make tank temperatures jump erractically.
These are the fish in my 20 gallon so as you have a guide:
4 green cories
3 blackskirt (black widow) tetras
2 bloodfin tetras
2 clown pleco
2 long fin blue danio
1 green tiger barb
1 rainbow shark
A plethora of mud snails.
My betta also does not have a heater.
~~Colesea