The aircraft checklist will determine when the pitot heat should be on. Strictly speaking, if the checklist says it should be on, the only reason it should not be on is if the system is inoperative or is causing some issue (if the generator system fails it will drain the battery, as falstro commented). The checklist linked by Lnafziger has the pitot heat ON before takeoff and OFF after landing. Other times it may be "as required" by conditions.
The pitot heat makes sure the pitot system remains free of ice. Failed airspeed, especially in IFR conditions, can be serious. The failure may not be obvious, leaving you to stall/overspeed. Leaving the heat on can cause the system to overheat. While some systems automatically protect against this, not all will (especially with smaller aircraft).
Checklists for larger jets also seem to have probe heat on before taxi and off right before shutdown. The reason for this is probably safety. Modern aircraft rely heavily on pitot static systems. AF447 crashed partially due to icing on the pitot tubes. Why leave it off and run the risk of icing? The cost of leaving it on is fairly low.
On smaller aircraft where the pilot may even be able to see the pitot tubes, the situation may be different. These aircraft have less automation, and the cost of repairing a pitot heat system is higher relative to the rest of the aircraft's maintenance. However, the issue of safety remains.