The resistor constantly produces heat with rate of $IR^2$; however, not all of this energy remains in the resistor (in form of internal thermal energy, ie. temperature).
So there's a sink: heat exchange between the resistor and environment (by radiation, conduction and convection).
Total change in temperature relates to net effect of the two. If the resistor constantly produces heat with some rate, but it also radiates energy with the same rate to the environment, the temperature basically remains constant, and this is what happens when you let a resistor go for a while.
EDIT (More Explanation):
Let's see what happens to a light bulb when you turn it up: Lamps are resistors (incandescent lamps).
When a lamp is off, heat production rate due to electricity is 0, and heat radiation rate is 0 too.
Just after you light it up, it begins to produce electrical energy with a rate of, say $100$ $J/s$ while heat radiation is still low, say $10$ $J/s$ because the lamp is still cold. This means that every second, $90$ $J$ of energy becomes internal energy and raises temperature of the lamp.
After a while, lamp gets warm. Warm objects release heat faster, say $60$ $J/s$. But it's still gaining heat with the $100$ $J/s$ rate. Therefore it's still warming up, but a bit slower; $40$ $J/s$.
When lamp gets hot enough so that it radiates energy with $100$ $J/s$ rate, since this is equal to electrical heat produced, these factors cancel out, building a balanced condition called equilibrium. At this moment, temperature no longer changes; even tough the heat is being produced all the time.